Convergence of the scalar- and vector-valued Allen–Cahn equation to mean curvature flow with 90°-contact angle in higher dimensions,part I: Convergence result

Abstract

We consider the sharp interface limit for the scalar-valued and vector-valued Allen–Cahn equation with homogeneous Neumann boundary condition in a bounded smooth domain Ω of arbitrary dimension $N ⩾ 2$ in the situation when a two-phase diffuse interface has developed and intersects the boundary $\partial Ω$ . The limit problem is mean curvature flow with 90°-contact angle and we show convergence in strong norms for well-prepared initial data as long as a smooth solution to the limit problem exists. To this end we assume that the limit problem has a smooth solution on $[0, T]$ for some time $T > 0$ . Based on the latter we construct suitable curvilinear coordinates and set up an asymptotic expansion for the scalar-valued and the vector-valued Allen–Cahn equation. In order to estimate the difference of the exact and approximate solutions with a Gronwall-type argument, a spectral estimate for the linearized Allen–Cahn operator in both cases is required. The latter will be shown in a separate paper, cf. (Moser (2021)).

Keywords

Sharp interface limit mean curvature flow contact angle Allen–Cahn equation vector-valued Allen–Cahn equation

1. Introduction

In the following we introduce the Allen–Cahn equation and the vector-valued variant considered in this paper. Moreover, we motivate and review results concerning the corresponding sharp interface limits.

Let us begin with the scalar Allen–Cahn equation. Let $N \in N$ , $N ⩾ 2$ and $Ω \subset R^{N}$ be a bounded, smooth domain with outer unit normal $N_{\partial Ω}$ . Moreover, let $ε > 0$ small. For $T > 0$ and $u_{ε} : \overline{Ω} \times [0, T] \to R$ we consider the Allen–Cahn equation with homogeneous Neumann boundary condition (AC) consisting of $\begin{array}{l} (AC1) & \partial_{t} u_{ε} - Δ u_{ε} + \frac{1}{ε^{2}} f^{'} (u_{ε}) = 0 in Ω \times (0, T), \\ (AC2) & \partial_{N_{\partial Ω}} u_{ε} = 0 on \partial Ω \times (0, T), \\ (AC3) & u_{ε} |_{t = 0} = u_{0, ε} in Ω, \end{array}$ where $f : R \to R$ is a suitable smooth double well potential with wells of equal depth. A typical example is $f (u) = \frac{1}{2} {(1 - u^{2})}^{2}$ , see Fig. 1. The precise conditions are $\begin{matrix} (1.1) & \begin{matrix} f \in C^{\infty} (R), f^{'} (\pm 1) = 0, f^{″} (\pm 1) > 0, \\ f (- 1) = f (1), f > f (1) in (- 1, 1) \end{matrix} \end{matrix}$ and we assume $\begin{matrix} (1.2) & u f^{'} (u) ⩾ 0 for all | u | ⩾ R_{0} and some R_{0} ⩾ 1 . \end{matrix}$ Note that (1.2) is just a requirement for the sign of $f^{'}$ outside a large ball. The condition (1.2) is used later to obtain uniform a priori bounds for classical solutions $u_{ε}$ , see Section 6.1.1 below.

Fig. 1.

Typical form of the double-well potential, $f (u) = \frac{1}{2} {(1 - u^{2})}^{2}$ .

The Allen–Cahn equation (similar to (AC1)) was originally introduced by Allen and Cahn [11] to describe the evolution of antiphase boundaries in certain polycrystalline materials. Moreover, one can directly verify that equation (AC1)–(AC3) is the $L^{2}$ -gradient flow to the energy $\begin{matrix} (1.3) & E_{ε} (u) : = \int_{Ω} \frac{1}{2} | \nabla u |^{2} + \frac{1}{ε^{2}} f (u) d x . \end{matrix}$ The latter is (up to a scaling in ε) the usual scalar Ginzburg–Landau energy or Modica–Mortola energy, cf. Modica [65]. For a summary of further motivations we refer to the introduction in Bronsard, Reitich [23].

The Allen–Cahn equation is a diffuse interface model: the $u_{ε}$ serves as an order parameter, where the values $\pm 1$ correspond to two distinct phases in applications. Typically after a short time Ω is partitioned into subdomains where the solution $u_{ε}$ of (AC1)–(AC3) is close to $\pm 1$ and transition zones (diffuse interfaces; roughly $u_{ε}^{- 1} ([- 1 + μ, 1 - μ])$ for $μ > 0$ small) develop where $| \nabla u_{ε} |$ is large. See Fig. 2 below for a typical situation. For a rigorous result in this direction (“generation of interfaces”) see Chen [27]. Formally, one can see this in the equation since the “reaction term” $f^{'} (u_{ε}) / ε^{2}$ should be large for small times away from the minima of f compared to the diffusion term $Δ u_{ε}$ . One also speaks of fast reaction/slow diffusion, see Rubinstein, Sternberg, Keller [72]. Neglecting $Δ u_{ε}$ , (AC1) becomes an ODE in time for each space point. For this ODE the stationary points are 0, $\pm 1$ , where 0 is unstable and $\pm 1$ are stable. Moreover, one can also have a look at the energy (1.3). It is well-known that solutions to the corresponding gradient flow behave in such a way that the energy is non-increasing (and decreases in some optimal sense) in time. In this perspective, values of $u_{ε}$ away from $\pm 1$ are penalized strongly and values of $| \nabla u_{ε} |$ away from 0 are penalized weakly. Heuristically (or in sufficiently smooth cases) one can argue that the thickness of the diffuse interfaces is proportional to ε. Hence for $ε \to 0$ one should obtain a hypersurface $Γ = {(Γ_{t})}_{t \in [0, T]}$ evolving in time, cf. Fig. 2, and hence a sharp interface model. Such limits are therefore called “sharp interface limits”.

Fig. 2.

Diffuse interface and sharp interface limit.

Next, let us introduce the vector-valued Allen–Cahn equation considered in the paper. Let N, Ω, $N_{\partial Ω}$ , ε, T be as above. Moreover, let $m \in N$ and $W : R^{m} \to R$ be a suitable potential which will be specified below. Let $ε > 0$ be a small parameter. Then for ${\vec{u}}_{ε} : \overline{Ω} \times [0, T] \to R^{m}$ we consider the vector-valued Allen–Cahn equation with Neumann boundary condition (vAC) consisting of $\begin{array}{l} (vAC1) & \partial_{t} {\vec{u}}_{ε} - Δ {\vec{u}}_{ε} + \frac{1}{ε^{2}} \nabla W ({\vec{u}}_{ε}) = 0 in Ω \times (0, T), \\ (vAC2) & \partial_{N_{\partial Ω}} {\vec{u}}_{ε} = 0 on \partial Ω \times (0, T), \\ (vAC3) & {\vec{u}}_{ε} |_{t = 0} = {\vec{u}}_{0, ε} in Ω . \end{array}$

In analogy to the scalar case one can compute directly that equation (vAC1)–(vAC3) is the $L^{2}$ -gradient flow to the vector-Ginzburg–Landau energy $\begin{matrix} (1.4) & {\overset{ˇ}{E}}_{ε} (\vec{u}) : = \int_{Ω} \frac{1}{2} | \nabla \vec{u} |^{2} + \frac{1}{ε^{2}} W (\vec{u}) d x . \end{matrix}$ See also Bronsard, Reitich [23] for further motivations.

The potential W. Here we allow two types of potentials W. On the one hand, we consider W with exactly two distinct minima and symmetry with respect to the hyperplane in the middle of these. On the other hand, we consider $m = 2$ and triple-well potentials W with symmetry. The first type is basically only interesting from a technical point of view, where with the second type one can describe e.g. three distinct phases in a polycristalline material, cf. [23]. The precise requirements are as follows:

Fig. 3.

Typical triple-well potential W. The image is taken from Kusche [54].

Definition 1.1.

Let $W : R^{m} \to R$ be smooth and one of the two assumptions below hold. In both cases we additionally require $\vec{u} \cdot \nabla W (\vec{u}) ⩾ 0$ for all $\vec{u} \in R^{m}$ , $| \vec{u} | ⩾ {\overset{ˇ}{R}}_{0}$ for an ${\overset{ˇ}{R}}_{0} > 0$ . Moreover, we assume that the kernel to a certain linear operator associated to W is one-dimensional, see Remark 4.11 below for the precise condition and an explanation. The two cases are:

W has exactly two global minima $\vec{a}, \vec{b}$ with $W (\vec{a}) = W (\vec{b}) = 0$ in which $D^{2} W$ is positive definite and W is symmetric with respect to the reflection $R_{\vec{a}, \vec{b}} : R^{m} \to R^{m}$ at the hyperplane $\frac{1}{2} (\vec{a} + \vec{b}) + span {\vec{a} - \vec{b}}^{⊥}$ .

W is a symmetric triple well-potential for $m = 2$ , i.e. W has exactly three global minima ${\vec{x}}_{i}$ , $i = 1, 3, 5$ with $W ({\vec{x}}_{i}) = 0$ for $i = 1, 3, 5$ in which $D^{2} W$ is positive definite and W is symmetric with respect to the symmetry group G of the equilateral triangle, cf. Kusche [54, Section 3.2] for the precise definition of G. Note that G consists of the three reflections at the symmetry axes and the rotations by $0, 120$ and 240 degrees.

Remark 1.2.

An example for a typical triple-well potential that fulfils the conditions in Definition 1.1 can be found in [54, Section 3.4]. See Fig. 3 above. The example stems from Haas [45], see also Garcke, Haas [43].

Compared to the scalar case, we always require symmetry properties for the potential W. The assumption is used e.g. in Section 4.3.1 below, but it might be possible to relax this.

Analogously to the scalar case one can argue with formal arguments (system of fast reaction and slow diffusion; gradient flow to the energy (1.4)) that diffuse interfaces for solutions of (vAC) should develop after short time. Note that the transition between minima of W runs in $R^{m}$ . Moreover, in the case of a triple-well potential W also three-fold diffuse interfaces between the three minima of W are possible.

Sharp/diffuse interface models and sharp interface limits. In general, sharp interface models and diffuse interface models are important model categories for the description of interfaces and moving boundaries in a large variety of applications. Some prominent examples are the melting of ice, the motion of an oil droplet in water, crystal growth, biological membranes, porous media, tumour evolution, spinodal decomposition of polymers and grain boundaries, see e.g. [11,14,15,19,33,34,42,63,68,70] and the references therein.

It is an important task to connect diffuse interface models and sharp interface models via their sharp interface limits for the following reasons (see also the partly universal introduction in Caginalp, Chen [24] and general comments in Caginalp, Chen, Eck [25]):

Modelling and Analysis: both types of models can usually be derived or motivated with physical principles, phenomenological observations or geometrical arguments etc., but one always incorporates some constitutive assumptions. Often the derivation for the sharp interface models is more transparent and these models appear simpler and more qualitative. On the other hand, diffuse interface models are usually advantageous in more complicated situations (see e.g. [14], p. 141) and solutions typically have better analytical properties. Especially topology changes do not impose any difficulties. By identifying the sharp interface limit one confirms that the assumptions in the derivations are appropriate as well as that the models are compatible with each other and can be used to describe the same situation. Another motivation is the concept of using the diffuse interface model to extend solutions of the corresponding sharp interface model past singularities.

Numerics: diffuse interface models are often simpler to solve numerically. By considering the sharp interface limit one justifies that the numerical solution to the diffuse interface model can be used to approximate the solution to the sharp interface model.

Concerning results for sharp interface limits: in general there are formal results and rigorous proofs for convergence. The formal sharp interface limits are typically based on formal asymptotic expansions or numerical experiments. However, see also Anderson, McFadden, Wheeler [14], p. 156ff for a “pillbox argument”, i.e. reasoning with a small test volume. Regarding rigorous sharp interface limits, one can basically group such results into two types:

Local time results that are applicable before singularities appear, i.e. as long as the interface does not develop singularities and stays smooth. Relatively “strong” results are obtained, e.g. norm estimates.

Global time results using some kind of weak notion for the sharp interface system, e.g. viscosity solutions for mean curvature flow, varifold solutions, distributional solutions, etc.

Formal and rigorous sharp interface limit for the Allen–Cahn-equation. In the situation of the scalar Allen–Cahn equation (AC) formal asymptotic analysis by Rubinstein, Sternberg, Keller [72] yields that the limit sharp interface ${(Γ_{t})}_{t \in [0, T]}$ should evolve according to the mean curvature flow $\begin{matrix} (MCF) & V_{Γ_{t}} = H_{Γ_{t}} \end{matrix}$ and, if there is boundary contact, there should be a 90°-contact angle. Moreover, the numerical experiments in Lee, Kim [57] give another confirmation on a formal level for the convergence of (AC) to (MCF) with 90°-contact angle. Finally, note that the 1D-case is not interesting for finite time in the t-scale because patterns persist for and evolve in exponentially slow time scales $τ = e^{- c / ε} t$ , cf. Carr, Pego [26]. More precisely, the sharp interface is a point and does not move in the time-scale t. This is consistent with (MCF) when the curvature of a point is defined as zero.

For the vector-valued Allen–Cahn equation (vAC) formal asymptotic calculations in Bronsard, Reitich [23] yield that for a triple-well potential W in the sharp interface limit $ε \to 0$ one should obtain (MCF) together with:

A 90°-contact angle if a transition of two phases meets the boundary.

A 120°-triple junction if the three phases meet at an interior point.

For the sake of completeness, note that the limit $ε \to 0$ in energies of the form (1.3) (with similar potentials) has been considered in the context of Γ-convergence,1

Of course this “Γ” has a different meaning than the Γ in Fig. 2.

see Modica [65] (with mass constraint) and Sternberg [76] (with and without mass constraint). The Γ-limits are perimeter functionals which (at least formally) induce (MCF) with 90°-contact angle via the

L^{2}

-gradient flow. This also motivates to study the dynamical problem (AC1)–(AC3) associated to the energy (1.3) and its relation to (MCF) with 90°-contact angle in the limit

ε \to 0

For results in the direction of Γ-convergence with respect to $ε \to 0$ for energies like (1.4) (for several types of potentials and usually with mass constraint) see Baldo [16] and the references therein. The Γ-limits are (multiphase) perimeter functionals which (at least formally) induce (multiphase) mean curvature flow via the $L^{2}$ -gradient flow. This gives another motivation to study the dynamical problem (vAC1)–(vAC3) and the connection to (multiphase) mean curvature flow in the limit $ε \to 0$ .

There are many rigorous results on the sharp interface limit for the Allen–Cahn equation ((AC1) on $R^{N}$ or (AC)) to (MCF) (in the case of (AC) with 90°-contact angle).

We start with the local time results. Via a comparison principle and the construction of sub- and supersolutions, Chen [27] proves local in time convergence as long as the interface stays smooth. Moreover, de Mottoni and Schatzman [30] consider the $R^{N}$ -case and show convergence with strong norms for times when a smooth solution to (MCF) exists by using rigorous asymptotic expansions and a spectral estimate for an associated linear operator. This method (which we will refer to as the “de Mottoni Schatzman method” or the “expansions method”) will be used in the paper and is explained in detail below. This also works for (AC) when the interface is closed and strictly contained in Ω. Note that the papers by Chen, Hilhorst, Logak [29] and Abels, Liu [2] also yield results for (AC) with strictly contained interface by simple adjustments. The resulting proofs and results use ideas from [30] but are more optimized. In Abels, Moser [5] the methods in [2,29,30] are generalized to the case of boundary contact of the diffuse interface in two dimensions. Moreover, there is the result by Fischer, Laux, Simon [40], where a relative entropy method is used. Finally, note that there is a paper by Sáez [73], but unfortunately there is a severe gap in the proof of the main theorem, cf. [5] for details.

For global time results one has to use some weak formulation of (MCF). There is the notion of viscosity solutions used by Evans, Soner, Souganidis [37] for $Ω = R^{N}$ and by Katsoulakis, Kossioris, Reitich [52] in the case of a convex, bounded domain. In the latter the maximum principle is used and sub- and supersolutions to the Allen–Cahn equation are constructed using the distance function from the level set of a viscosity solution for (MCF). For another general approach in this direction cf. Barles, Souganidis [18] and Barles, Da Lio [17]. The latter are more geometrical and the convexity assumption for the domain is not needed, but one relies on geometric maximum principles. Moreover, varifold solutions to (MCF) are used by Ilmanen [49] in the $R^{N}$ -case, by Mizuno, Tonegawa [64] for smooth, strictly convex, bounded domains and by Kagaya [50] without the convexity assumption. For varifold solutions only convergence of a subsequence is achieved. Finally, there is the conditional result by Laux, Simon [56] where convergence of the (vector-valued; scalar case contained) Allen–Cahn equation to (multiphase) mean curvature flow in a BV-setting is obtained.

Altogether, there is a large variety of results. At the time of writing, [27] and [5] were the only results of local type that allow boundary contact for the diffuse interfaces. Moreover, at the time of writing there was only the conditional result [56] in the vector-valued case on the convergence of the vector-valued Allen–Cahn equation to multiphase mean curvature flow in a BV-setting. But very recently, the relative entropy approach from [40] has been generalized to the vector-valued case by Fischer, Marveggio [41], even for potentials with more than three wells, but at the moment for a slightly different class of potentials compared to Definition 1.1 above, e.g. not allowing $C^{2}$ , but $C^{1, 1}$ -regularity. The idea for the latter method is to define suitable energies that control the error of the diffuse and sharp interface solution. Then Gronwall-type estimates are derived for these energies. The method itself originates to results on conservation laws and was recently extended to curvature driven interface evolution problems resulting into many results of weak-strong uniqueness type, cf. for example [39] and see the references in [40,41]. Then the relative entropy method was adjusted for sharp interface limits in [40]. In this context let us mention Laux, Liu [55] who study nematic–isotropic phase transition in liquid crystals via the relative entropy (or modulated energy) method. Moreover, note the paper by Liu [59] on the sharp interface limit of an anisotropic Ginzburg–Landau model obtained with this method. Finally, there is the work Hensel, Moser [47] who consider formally the same system (but with the relative entropy method) as in [6] where the limit problem is mean curvature flow with constant contact angle. In [47] the angle is allowed to be arbitrary in comparison to [6], but the assumptions on a certain boundary energy density are in fact disjoint. Roughly speaking, [6] relies rather on smoothness assumptions and perturbation arguments, whereas some kind of positivity for the energy estimates in [47] is needed. The situation seems to be similar with [41]. Summarizing the above, the relative entropy approach for sharp interface limits turned out to be more flexible for lower regularities and for more complicated topologies compared to the expansions method. Moreover, it does not rely on spectral estimates. However, the detailed profile of the diffuse interface is (at the moment) not obtained compared to the expansions method and e.g. treating different viscosities with the relative entropy method is more difficult, compare [39] and [1]. In [39] the weak-strong uniqueness for the two-phase Navier–Stokes equation with the relative entropy method is shown, and the error of a varifold and a strong solution is estimated. In [1] the sharp interface limit for a Navier–Stokes/Allen–Cahn system with different viscosities is considered.

Note that there is a work by Sáez [74] on the vector-valued problem, but unfortunately there is the same gap in the proof as in [73], cf. [5] for details. It is difficult to generalize the methods in [27] because comparison principles are used. On the other hand, the method by de Mottoni and Schatzman [30] has also proven to be versatile and was applied to many other diffuse interface models as well, see the comments below.

This motivates to extend the result in [5] to more complicated geometrical situations and equations. In this paper we generalize the latter in two directions. On the one hand, we look at the higher-dimensional setting and on the other hand, we also consider the situation of a two-phase transition for the vector-valued Allen–Cahn equation. The following results are obtained:

Convergence of (solutions to) the scalar-valued Allen–Cahn equation (AC) to (MCF) with 90°-contact angle in any dimension $N ⩾ 2$ . See Section 1.1 below. For the case $N = 2$ see also [5].

Convergence of (solutions to) the vector-valued Allen–Cahn equation (vAC) to (MCF) with 90°-contact angle in any dimension $N ⩾ 2$ , see Section 1.2 below. Here we only treat the case of two-phase transitions, i.e. triple junctions are excluded. The result is an important building block for the triple junction case, since the arguments are localizable.

The results are part of the PhD thesis of the author, cf. Moser [66]. In a separate paper by Abels, Moser [6] the work [5] is extended for the case of a non-linear Robin boundary condition which is designed in such a way that in the sharp interface limit (MCF) with α-contact angle is attained, where α is fixed but close to 90°.

The method of de Mottoni and Schatzman. One assumes that there exists a local smooth solution to the limit sharp interface problem. This can typically be shown for small times. Then

One rigorously constructs an approximate solution to the diffuse interface model using asymptotic expansions based on the evolving surface that is (part of) the solution to the limit problem. To this end one has to solve model problems for the series coefficients.

Then one estimates the difference between exact and approximate solutions with a Gronwall-type argument. This typically involves a spectral estimate for a linear operator associated to the diffuse interface equation and the approximate solution.

In the method comparison principles are not needed in a fundamental way compared to many other approaches. However, the application can be difficult for complicated equations or too singular situations. On the other hand, one gets a strong result: norm estimates and the typical profile of the solution are obtained.

Therefore the method was used for many other diffuse interface models as well. These results are based on general spectrum estimates in Chen [28] for Allen–Cahn, Cahn–Hilliard and phase-field-type operators. For the subtle variations in the rigorous asymptotic expansions and the spectral estimates used in applications of the method by de Mottoni and Schatzman see the inceptions to Sections 5 and 6 in [66]. There are results for the Cahn–Hilliard equation by Alikakos, Bates, Chen [9], the phase-field equations by Caginalp, Chen [24], the mass-conserving Allen–Cahn equation by Chen, Hilhorst, Logak [29], the Cahn–Larché system by Abels, Schaubeck [7] and a Stokes/Allen–Cahn system by Abels, Liu [2]. Moreover, there is the paper [1] on the sharp interface limit for a Navier–Stokes/Allen–Cahn system with different viscosities. Moreover, Marquardt [62] (see also [3,4]) studied the sharp interface limit for a Stokes/Cahn–Hilliard system. See also Schaubeck [75] for a result on a convective Cahn–Hilliard equation. Furthermore, there is the result by Fei, Liu [38], where a phase field approximation for the Willmore flow is considered. Finally, [5] is the first result obtained with the method of de Mottoni and Schatzman that allows boundary contact for the diffuse interfaces.

Structure of the paper. In Sections 1.1–1.2 we formulate the main results for the scalar-valued and vector-valued Allen–Cahn equation, (AC) and (vAC) respectively. In Section 2 we fix some notation and introduce function spaces. In Section 3 suitable curvilinear coordinates are constructed. In Section 4 we solve the model problems appearing in the asymptotic expansions in Section 5. The difference estimates and the proofs of the convergence theorems are carried out in Section 6. Therefore suitable spectral estimates are needed, see [67] for those.

Note that Section 4 on the model problems can be read independently since it solely relies on Section 2 on function spaces. Moreover, Section 3 on the existence of curvilinear coordinates is also self-contained, and does only need some notation from the beginning of Section 2. Section 5 on the asymptotic expansions then needs the results of Sections 3–4, but just the definitions from Section 2. The main point is then the short Section 5.1.3 and 5.2.3, respectively. The latter can be understood without precisely reading the remainder of Section 5 and is the only part that is needed from Section 5 for Section 6. Finally, apart from that, Section 6 relies on Section 3 and the spectral estimates from [67].

Comparison to [ 5 ]. Compared to [5] the geometrical situation is more difficult because the dimension is arbitrary. However, with similar but more technical arguments we also obtain suitable curvilinear coordinates in Section 3. The model problems in the scalar case are the same as in [5], cf. Sections 4.1–4.2, but for the half space problem we use a new setting with exponentially weighted Sobolev spaces which allows for solution operators. With the latter the iteration of the asymptotic expansion in Section 5.1 is more efficient and one can solve the appearing model problems in [6] via the Implicit Function Theorem for angles α close to $\frac{π}{2}$ . In comparison with the 2D-situation in [5] the asymptotic expansion is similar but more technical due to the geometry and we also iterate it here. For instance, at the contact manifold, another independent variable appears. The difference estimate and the proof of the convergence result is also similar but more technical compared to [5], see Section 6.2.

In the vector-valued case, new ideas are required for the ODE model problems on the real line, cf. Section 4.3. For the nonlinear vector-valued ODE in Section 4.3.1 an energy minimization argument is used, which was done before by [22] and [54]. In order to solve the linearized vector-valued ODE in Section 4.3.3 we make the assumption that the kernel of the corresponding linearized operator is 1-dimensional, cf. Remark 4.11. This then gives rise to a solution operator for the linearized vector-valued ODE, also in exponentially weighted spaces via perturbation arguments. Moreover, the assumption in Remark 4.11 yields a spectral gap as in the scalar case, cf. Lemma 4.12. Based on this, one can solve the vector-valued model problem on the half space in the analogous way as in the scalar case, see Section 4.4. The asymptotic expansion in Section 5.2 and the difference estimate in Section 6.3 is then analogous to the scalar case. Finally, note that the spectral estimates needed for Section 6.3 also had to be adjusted suitably from the scalar case to the vector-valued case, cf. [67]. It was unclear a priori how to achieve the latter, because in the scalar case there was frequent use of theorems such as the maximum principle for scalar 1D-operators. This is another reason why the assumption on the kernel from Remark 4.11 is used.

Mean curvature flow ( MCF ) with 90°-contact angle, coordinates and notation. “Mean curvature flow” for evolving hypersurfaces means that the normal velocity equals mean curvature, where we define for convenience “mean curvature” as the sum of the principal curvatures.

For the convergence result below we will assume that (MCF) together with a 90°-contact angle condition at $\partial Ω$ has a smooth solution on a time interval $[0, T_{0}]$ . This is a prerequisite for the method of de Mottoni and Schatzman [30].

The local well-posedness and existence of a smooth solution for small time starting from suitable initial sharp interfaces is basically well-known. At this point let us give some references in this direction. In Katsoulakis, Kossioris, Reitich [52, Section 2], a parametric approach is used to show local existence and uniqueness of classical solutions for (MCF) in arbitrary dimension and with fixed contact angle. In principle, it is also possible to reduce the evolution to a parabolic PDE by writing it over a reference hypersurface via suitable coordinates. For the typical procedure in the case of a closed interface see Prüss, Simonett [70]. For curvilinear coordinates in the situation of boundary contact see Vogel [79] and Section 3 below. Moreover, note that in Huisken [48] the special case of (MCF) with 90°-contact angle in the graph case for cylindrical domains is considered and global existence and uniqueness of smooth solutions as well as convergence to a constant graph is obtained.

We need some notation in the context of the curvilinear coordinates in order to formulate the main theorems below.

Remark 1.3 (Domain, Sharp Interface and Coordinates).

For the details see Section 3.

Domain. Let $N \in N$ , $N ⩾ 2$ and $Ω \subset R^{N}$ be a bounded, smooth domain with outer unit normal $N_{\partial Ω}$ . For $T > 0$ we set $Q_{T} : = Ω \times (0, T)$ and $\partial Q_{T} : = \partial Ω \times [0, T]$ .

Sharp Interface. Consider $T_{0} > 0$ and an evolving hypersurface $Γ = {(Γ_{t})}_{t \in [0, T_{0}]}$ (with boundary, smooth, oriented, compact, connected) suitably parametrized over a reference hypersurface Σ and such that $\partial Γ$ meets $\partial Ω$ at contact angle $\frac{π}{2}$ . For Γ one can define the normal velocity $V_{Γ_{t}}$ and mean curvature $H_{Γ_{t}}$ at time $t \in [0, T_{0}]$ with respect to a unit normal $\vec{n}$ of Γ in the classical sense. See Section 3.1 for the precise assumptions and definitions.

Coordinates. We construct appropriate curvilinear coordinates $(r, s)$ with values in $[- 2 δ, 2 δ] \times Σ$ for some $δ > 0$ describing a neighbourhood of Γ in $\overline{Ω} \times [0, T_{0}]$ . For the exact statements see in particular Theorem 3.2, with $2 δ$ instead of δ there. Here r has the role of a signed distance function and s works like a tangential projection. The set $\overline{Q_{T_{0}}} = \overline{Ω} \times [0, T_{0}]$ is split by Γ into two connected sets (Γ excluded) according to the sign of r. We denote them with $Q_{T_{0}}^{\pm}$ . Then we have the disjoint union $\begin{matrix} \overline{Q_{T_{0}}} = Γ \cup Q_{T_{0}}^{-} \cup Q_{T_{0}}^{+} . \end{matrix}$ Finally, we introduce tubular neighbourhoods $Γ (η) : = r^{- 1} ((- η, η))$ for $η \in (0, 2 δ]$ and define a suitable normal derivative $\partial_{n}$ and tangential gradient $\nabla_{τ}$ on $Γ (η)$ , see Remark 3.3.

1.1. Result in scalar case

Finally, we state the new convergence result for (AC) obtained with the method of de Mottoni and Schatzman [30].

Theorem 1.4 (Convergence of (AC) to (MCF) with 90°-Contact Angle).

Let $N ⩾ 2$ , Ω, $N_{\partial Ω}$ , $Q_{T}$ and $\partial Q_{T}$ for $T > 0$ be as in Remark 1.3 , 1. Moreover, let $Γ = {(Γ_{t})}_{t \in [0, T_{0}]}$ for some $T_{0} > 0$ be a smooth evolving hypersurface with $\frac{π}{2}$ -contact angle condition as in Remark 1.3 , 2. and let Γ satisfy ( MCF ). Let $δ > 0$ small and the notation for $Q_{T_{0}}^{\pm}$ , $Γ (δ)$ , $\nabla_{τ}$ , $\partial_{n}$ be as in Remark 1.3 , 3. Moreover, let f satisfy ( 1.1 )–( 1.2 ). Let $M \in N$ with $M ⩾ k (N) : = max {2, \frac{N}{2}}$ .

Then there are $ε_{0} > 0$ and $u_{ε}^{A} : \overline{Ω} \times [0, T_{0}] \to R$ smooth for $ε \in (0, ε_{0}]$ (depending on M) with ${lim}_{ε \to 0} u_{ε}^{A} = \pm 1$ uniformly on compact subsets of $Q_{T_{0}}^{\pm}$ and such that the following holds:

If $M > k (N)$ , then let $u_{0, ε} \in C^{2} (\overline{Ω})$ with $\partial_{N_{\partial Ω}} u_{0, ε} = 0$ on $\partial Ω$ for $ε \in (0, ε_{0}]$ and $\begin{array}{l} (1.5) & sup_{ε \in (0, ε_{0}]} ‖ u_{0, ε} ‖_{L^{\infty} (Ω)} < \infty and {‖ u_{0, ε} - u_{ε}^{A} |_{t = 0} ‖}_{L^{2} (Ω)} ⩽ R ε^{M + \frac{1}{2}} \end{array}$ for some $R > 0$ and all $ε \in (0, ε_{0}]$ . Then for any set of solutions $u_{ε} \in C^{2} (\overline{Q_{T_{0}}})$ of ( AC1 )–( AC3 ) for $ε \in (0, ε_{0}]$ with initial values $u_{0, ε}$ there are $ε_{1} \in (0, ε_{0}]$ , $C > 0$ such that $\begin{matrix} (1.6) & \begin{matrix} sup_{t \in [0, T]} {‖ (u_{ε} - u_{ε}^{A}) (t) ‖}_{L^{2} (Ω)} + {‖ \nabla (u_{ε} - u_{ε}^{A}) ‖}_{L^{2} (Q_{T} ∖ Γ (δ))} ⩽ C ε^{M + \frac{1}{2}}, \\ {‖ \nabla_{τ} (u_{ε} - u_{ε}^{A}) ‖}_{L^{2} (Q_{T} \cap Γ (δ))} + ε {‖ \partial_{n} (u_{ε} - u_{ε}^{A}) ‖}_{L^{2} (Q_{T} \cap Γ (δ))} ⩽ C ε^{M + \frac{1}{2}} \end{matrix} \end{matrix}$ for all $ε \in (0, ε_{1}]$ and $T \in (0, T_{0}]$ .

If $k (N) \in N$ and $M ⩾ k (N) + 1$ , then there is a $\tilde{R} > 0$ small such that the assertion in 1. holds, when $R, M$ in ( 1.5 )–( 1.6 ) are replaced by $\tilde{R}, k (N)$ .

If $N \in {2, 3}$ and $M = 2 (= k (N))$ , then there is $T_{1} \in (0, T_{0}]$ such that the conclusion in 1. is valid but only with $T \in (0, T_{1}]$ in ( 1.6 ).

Remark 1.5.

Interpretation of Theorem 1.4 . One can interpret $u_{ε}^{A}$ in the theorem as representation of a diffuse interface moving with Γ since $u_{ε}^{A}$ is smooth but converges for $ε \to 0$ to a step function whose jump set is the solution Γ to (MCF) with $\frac{π}{2}$ -contact angle starting from $Γ_{0}$ . The assumption on the initial values $u_{0, ε}$ in Theorem 1.4 essentially means that a diffuse interface already has developed and is located at the initial sharp interface $Γ_{0}$ at time $t = 0$ , i.e. the generation of diffuse interfaces in the evolution is skipped. One also speaks of “well-prepared initial data”, cf. [2]. Hence Theorem 1.4 basically shows that the qualitative behaviour of diffuse interfaces with boundary contact, generated by (AC), is that of (MCF) with 90°-contact angle, at least as long as the evolution of the latter stays smooth.

Layout of the Proof. Required model problems, some ODEs on $R$ and a linear elliptic equation on $R \times (0, \infty)$ are considered in Section 4.1 and Section 4.2 below, respectively. The asymptotic expansions are carried out in Section 5.1. The approximate solution $u_{ε}^{A}$ is defined in Section 5.1.3. Note that M corresponds to the number of terms in the expansion. The spectral estimate is proven in [67] and the difference estimate in Section 6.2.1. Finally, Theorem 1.4 is obtained in Section 6.2.2.

Well-Posedness of (AC). In Theorem 1.4 the existence of solutions $u_{ε} \in C^{2} (\overline{Q_{T_{0}}})$ of (AC) is assumed, but this is in principle well-known, see the references in Bellettini [21, Remark 15.1]. An approach with weak solutions (obtained via time-discretization) can also be found in Bartels [20, Chapter 6.1]. Moreover, equation (AC1)–(AC3) fits in the general framework of Lunardi [60, Section 7.3.1], where a semigroup approach and a Hölder-setting is used. Together with a priori boundedness of classical solutions (see Section 6.1.1 below) that can be obtained with maximum principle arguments, one can show global well-posedness for regular, bounded initial data. See e.g. [60, Proposition 7.3.2]. Higher regularity then follows using linear theory, cf. Lunardi, Sinestrari, von Wahl [61]. Finally, note that well-posedness for (AC1) on $R^{N}$ is shown in [30] for bounded initial data with estimates for the heat semigroup.

The approximate solution $u_{ε}^{A}$ obtained from the explicit construction equals $\pm 1$ in the set $Q_{T_{0}}^{\pm} ∖ Γ (2 δ)$ and has a smooth, increasingly steep transition with a known “optimal profile” inbetween. Therefore Theorem 1.4 also yields the typical profile of solutions to (AC) across diffuse interfaces.

The level sets ${u_{ε}^{A} = 0}$ , ${u_{ε} = 0}$ can be viewed as approximations for Γ. Note that in the explicit construction of $u_{ε}^{A}$ in Section 5.1 below the error from ${u_{ε}^{A} = 0}$ to Γ is of order ε and, in the case that f is even, of order $ε^{2}$ , see Remark 5.9. If one uses numerical computations for (AC) in order to approximate solutions to (MCF) with 90°-contact angle condition this is of interest, cf. also Caginalp, Chen, Eck [25]. Note that for small ε, the set ${u_{ε}^{A} = 0}$ should actually give a proper surface, while for ${u_{ε} = 0}$ this will not be the case in general. However, if the initial data are good enough and the approximation error small in sufficiently strong norms, it should be possible to achieve this also for ${u_{ε} = 0}$ . For estimates in higher norms see the next item.

In principle also estimates of stronger norms (e.g. Hölder norms of arbitrary order) are possible in the situation of Theorem 1.4, but better estimates for the initial values could be required. The basic idea is to interpolate the already controlled norms with stronger norms that can be estimated for exact solutions by some negative ε-orders. Cf. Alikakos, Bates, Chen [9, Theorem 2.3] for a similar idea in the case of the Cahn–Hilliard equation. However, note that this does not improve the approximation of Γ in the sense of 5.

Theorem 1.4 and the above comments hold analogously for closed Γ moving by (MCF) and compactly contained in Ω. The proof is basically contained since the constructions are localizable.

1.2. Result in vector-valued case

Theorem 1.6 (Convergence of (vAC) to (MCF) with 90°-Contact Angle).

Let $N ⩾ 2$ , Ω, $N_{\partial Ω}$ , $Q_{T}$ and $\partial Q_{T}$ be as in Remark 1.3 , 1. Moreover, let $Γ = {(Γ_{t})}_{t \in [0, T_{0}]}$ for some $T_{0} > 0$ be a smooth evolving hypersurface with $\frac{π}{2}$ -contact angle condition as in Remark 1.3 , 2. and let Γ satisfy ( MCF ). Let $δ > 0$ be small and the notation for $Q_{T_{0}}^{\pm}$ , $Γ (δ)$ , $\nabla_{τ}$ , $\partial_{n}$ be as in Remark 1.3 , 3. Moreover, let $W : R^{m} \to R$ be as in Definition 1.1 and ${\vec{u}}_{\pm}$ be any distinct pair of minimizers of W. Finally, let $M \in N$ with $M ⩾ k (N) : = max {2, \frac{N}{2}}$ .

Then there are ${\overset{ˇ}{ε}}_{0} > 0$ and ${\vec{u}}_{ε}^{A} : \overline{Ω} \times [0, T_{0}] \to R^{m}$ smooth for $ε \in (0, {\overset{ˇ}{ε}}_{0}]$ (depending on M) with ${lim}_{ε \to 0} {\vec{u}}_{ε}^{A} = {\vec{u}}_{\pm}$ uniformly on compact subsets of $Q_{T_{0}}^{\pm}$ and such that the following holds:

If $M > k (N)$ , then let ${\vec{u}}_{0, ε} \in C^{2} {(\overline{Ω})}^{m}$ with $\partial_{N_{\partial Ω}} {\vec{u}}_{0, ε} = 0$ on $\partial Ω$ for $ε \in (0, {\overset{ˇ}{ε}}_{0}]$ and $\begin{matrix} (1.7) & sup_{ε \in (0, {\overset{ˇ}{ε}}_{0}]} ‖ {\vec{u}}_{0, ε} ‖_{L^{\infty} {(Ω)}^{m}} < \infty and {‖ {\vec{u}}_{0, ε} - {\vec{u}}_{ε}^{A} |_{t = 0} ‖}_{L^{2} {(Ω)}^{m}} ⩽ R ε^{M + \frac{1}{2}} \end{matrix}$ for some $R > 0$ and all $ε \in (0, {\overset{ˇ}{ε}}_{0}]$ . Then for any set of solutions ${\vec{u}}_{ε} \in C^{2} {(\overline{Q_{T_{0}}})}^{m}$ of ( vAC1 )–( vAC3 ) for $ε \in (0, {\overset{ˇ}{ε}}_{0}]$ with initial values ${\vec{u}}_{0, ε}$ there are ${\overset{ˇ}{ε}}_{1} \in (0, {\overset{ˇ}{ε}}_{0}]$ , $C > 0$ with $\begin{matrix} (1.8) & \begin{matrix} sup_{t \in [0, T]} {‖ ({\vec{u}}_{ε} - {\vec{u}}_{ε}^{A}) (t) ‖}_{L^{2} {(Ω)}^{m}} + {‖ \nabla ({\vec{u}}_{ε} - {\vec{u}}_{ε}^{A}) ‖}_{L^{2} {(Q_{T} ∖ Γ (δ))}^{N \times m}} ⩽ C ε^{M + \frac{1}{2}}, \\ {‖ \nabla_{τ} ({\vec{u}}_{ε} - {\vec{u}}_{ε}^{A}) ‖}_{L^{2} {(Q_{T} \cap Γ (δ))}^{N \times m}} + ε {‖ \partial_{n} ({\vec{u}}_{ε} - {\vec{u}}_{ε}^{A}) ‖}_{L^{2} {(Q_{T} \cap Γ (δ))}^{m}} ⩽ C ε^{M + \frac{1}{2}} \end{matrix} \end{matrix}$ for all $ε \in (0, {\overset{ˇ}{ε}}_{1}]$ and $T \in (0, T_{0}]$ .

If $k (N) \in N$ and $M ⩾ k (N) + 1$ , then there is a $\overset{ˇ}{R} > 0$ small such that the assertion in 1. holds, when $R, M$ in ( 1.7 )–( 1.8 ) are replaced by $\overset{ˇ}{R}, k (N)$ .

If $N \in {2, 3}$ and $M = 2 (= k (N))$ , then there is ${\overset{ˇ}{T}}_{1} \in (0, T_{0}]$ such that the conclusion in 1. is valid but only but only with $T \in (0, T_{1}]$ in ( 1.8 ).

Remark 1.7.

The interpretation of Theorem 1.6 is analogous to the one of Theorem 1.4, where convergence of (AC) to (MCF) with 90°-contact angle is obtained, cf. Remark 1.5, 1.

Layout of the Proof. The new model problems, some vector-valued ODEs on $R$ and a vector-valued linear elliptic equation on $R \times (0, \infty)$ are solved in Section 4.3 and Section 4.4 below, respectively. The asymptotic expansions are done in Section 5.2 and the approximate solution ${\vec{u}}_{ε}^{A}$ is defined in Section 5.2.3. Note that M corresponds to the number of terms in the expansion. The spectral estimate is shown in [67] and the difference estimate is proven in Section 6.3.1. Finally, Theorem 1.6 is obtained in Section 6.3.2.

Well-Posedness of (vAC). In general the analysis of systems is more challenging than that of single equations. However, the derivatives in (vAC1)–(vAC3) are decoupled and hence some methods from the scalar case can also be used for (vAC), e.g. regularity theory. Equation (vAC1)–(vAC3) matches the general setting of Lunardi [60, Section 7.3.1], where a semigroup method and Hölder-spaces are used. Moreover, by reduction to a scalar equation and maximum principle arguments, one can obtain a priori boundedness of classical solutions, see Section 6.1.2 below. Hence global well-posedness for regular, bounded initial data follows. Higher regularity can be obtained with linear theory for scalar equations, cf. Lunardi, Sinestrari, von Wahl [61].

The comments for Theorem 1.4 in Remark 1.5, 4.–7. hold similarly. Here ${\vec{u}}_{\pm}$ has the role of $\pm 1$ and the order of the approximation of Γ in the spirit of Remark 1.5, 5. is $ε^{2}$ . The value zero from the scalar case for the level sets should be replaced by (part of) a hyperplane or hypersurface inbetween ${\vec{u}}_{\pm}$ , e.g. $\frac{1}{2} ({\vec{u}}_{+} + {\vec{u}}_{-}) + span {{\vec{u}}_{+} - {\vec{u}}_{-}}^{⊥}$ .

2. Notation and function spaces

Let $N$ be the natural numbers and $N_{0} : = N \cup {0}$ . The symbol $K$ stands for an element of ${R, C}$ . Furthermore, we denote $R_{+} : = (0, \infty)$ and with $R_{+}^{2} : = R \times (0, \infty)$ the half-space in the plane. Often functions or expressions $x \mapsto f (x)$ are abbreviated as $f (\cdot)$ when the variable is clear from the context. Moreover, the Euclidean norm in $R^{m}$ , $m \in N$ and the Frobenius norm in $R^{m \times n}$ , $m, n \in N$ are for convenience denoted by $| \cdot |$ . The symbol “ $\vec{}$ ” indicates a vector or a vector-valued function. Moreover, objects (e.g. vectors, operators and constants) that are associated to a vector-valued setting often get the addition “ $\overset{ˇ}{}$ ”. Furthermore, a subset $Ω \subseteq R^{n}$ , $n \in N$ is called “domain”, if Ω is open, nonempty and connected. Additionally, restrictions or evaluations of functions are often indicated by “ $|_{\cdot}$ ”. The differential operators ∇, div and $D^{2}$ are defined to act just on spatial variables. For convenience let D denote the Jacobian and ∇ the transpose. Let X be a set and Y a normed space. Then $B (X, Y) : = {f : X \to Y bounded}$ . Let $X, Y$ be normed spaces over $K$ . Then $L (X, Y)$ denotes the set of bounded linear operators $T : X \to Y$ . Finally, note that we use the usual constant convention, i.e. unimportant constants can change from line to line, but are independent from the free variables in the particular case.

Consider $n, k \in N$ , $γ \in (0, 1)$ and let $Ω \subseteq R^{n}$ be open and nonempty. Furthermore, let B be a Banach space over $K = R$ or $C$ . We use the standard notation $C^{k} (Ω, B)$ , $C^{k} (\overline{Ω}, B)$ , $C_{b}^{k} (Ω, B)$ , $C_{b}^{k} (\overline{Ω}, B)$ , $C^{k, γ} (\overline{Ω}, B)$ for continuous, k-times continuously differentiable and Hölder-continuous functions (and versions with bounded or continuously extendible functions and derivatives), cf. [66, Definition 2.1] for details. Moreover, let $C_{0}^{\infty} (Ω)$ be the set of $f \in C^{\infty} (Ω, R)$ with compact support $supp f \subseteq Ω$ . Finally, $C_{0}^{\infty} (\overline{Ω}) : = {f |_{\overline{Ω}} : f \in C_{0}^{\infty} (R^{n})}$ .

Let $(M, A, μ)$ be a σ-finite, complete measure space and B be a Banach space over $K = R$ or $C$ . Then one can define the notions of (μ- or strongly-) measurable and (Bochner-)integrable functions $f : M \to B$ and the Bochner(–Lebesgue)-Integral, see Amann, Escher [13, Chapter X] for the definitions and properties. In particular the Lebesgue-spaces $L^{p} (M, B)$ for $1 ⩽ p ⩽ \infty$ are defined. If $B = K$ and it is clear from the context if $K = R$ or $C$ , then we omit B in the notation. Let $(M, g)$ be a $C^{1}$ -Riemannian submanifold of $R^{n}$ with (or without) boundary. We denote with $\partial M$ the boundary (defined via charts) and with $M^{\circ} : = M ∖ \partial M$ the interior. Let $L_{M}$ be the Lebesgue σ-Algebra of M and $λ_{M}$ the Riemann–Lebesgue Volume Measure of M. See [13, Chapters XI, XII.1] for the definitions and properties. In particular $(M, L_{M}, λ_{M})$ is a σ-finite complete measure space (and satisfies many more properties). Therefore the Bochner-Integral and the Lebesgue spaces are defined. Let g be the Euclidean Metric and let M have dimension m. Then one can show that $λ_{M}$ coincides with the m-dimensional Hausdorff-measure $H^{m}$ on M. Cf. with Evans, Gariepy [36, Chapter 3.3.4]. Therefore in the application later we write $H^{m}$ instead of $λ_{M}$ for convenience.

Next we introduce the used Sobolev spaces on domains in $R^{n}$ .

Definition 2.1.

Let $Ω \subseteq R^{n}$ , $n \in N$ be open and nonempty. Moreover, let $k \in N_{0}$ , $1 ⩽ p ⩽ \infty$ and B be a Banach space. Then $W^{k, p} (Ω, B)$ are the usual Sobolev-spaces, where $W^{0, p} (Ω, B) : = L^{p} (Ω, B)$ . We also write $H^{k} (Ω, B)$ instead of $W^{k, 2} (Ω, B)$ . If $B = K$ and it is clear from the context if $K = R$ or $C$ , then we omit B in the notation.

Let $n \in N$ . Then $H^{β} (R^{n})$ for $β > 0$ are the well-known $L^{2}$ -Bessel-Potential spaces and $W^{k + μ, p} (R^{n})$ for $k \in N_{0}$ , $μ \in (0, 1)$ and $1 ⩽ p < \infty$ the Sobolev–Slobodeckij spaces.

For the definitions and properties of scalar-valued function spaces, in particular embeddings, interpolation results and trace theorems see Adams, Fournier [8], Alt [12], Leoni [58] and Triebel [77,78]. Many properties can be generalized to vector-valued function spaces over domains, see e.g. Kreuter [53] and the references therein.

Furthermore, we need Sobolev spaces on submanifolds of $R^{n}$ . Therefore let $(M, g)$ be a m-dimensional compact Riemannian submanifold of $R^{n}$ with (or without) boundary and class $C^{l}$ , where $l \in N \cup {\infty}$ , $l ⩾ 1$ . Let $U \subset M^{\circ}$ be open and nonempty. Moreover, let B be a Banach space. Then $L^{p} (U, B)$ is defined. We also need Sobolev spaces on U.
Definition 2.2.
Let $x_{j} : U_{j} \to V_{j} \subseteq [0, \infty) \times R^{m - 1}$ for $j = 1, \dots, N$ be charts of M and $W_{j}$ open in $[0, \infty) \times R^{m - 1}$ with $\overline{W_{j}} \subset V_{j}$ compact for $j = 1, \dots, N$ and $⋃_{j = 1}^{N} x_{j}^{- 1} (W_{j}) = M$ .
Then for $k \in N$ , $k ⩽ l$ and $1 ⩽ p < \infty$ we define the Sobolev spaces $\begin{matrix} W^{k, p} (U, B) : = {f \in L^{p} (U, B) : f \circ x_{j}^{- 1} |_{x_{j} (U \cap U_{j}) \cap W_{j}} \in W^{k, p} (x_{j} (U \cap U_{j}) \cap W_{j}, B) \forall j} . \end{matrix}$

For $f \in W^{1, p} (U, B)$ , $1 ⩽ p < \infty$ we define (in analogy to the scalar case) the surface gradient $\begin{matrix} [\nabla_{U} f] \circ x_{j}^{- 1} |_{x_{j} (U \cap U_{j}) \cap W_{j}} : = \sum_{r, s = 1}^{m} g^{r s} \circ x_{j}^{- 1} \partial_{y_{r}} [f \circ x_{j}^{- 1} |_{x_{j} (U \cap U_{j}) \cap W_{j}}] \partial_{y_{s}} (x_{j}^{- 1}) \end{matrix}$ for all $j = 1, \dots, N$ , where ${(g^{r s})}_{r, s = 1}^{m}$ is the inverse of the representation matrix of g with respect to $x_{j}$ and the product of $\partial_{y_{r}} [f \circ x_{j}^{- 1} |_{x_{j} (U \cap U_{j}) \cap W_{j}}] \in L^{p} (x_{j} (U \cap U_{j}) \cap W_{j}, B)$ with $\partial_{y_{s}} (x_{j}^{- 1}) \in C^{l} (\overline{x_{j} (U \cap U_{j}) \cap W_{j}}, R^{n})$ is understood component-wise.

Lemma 2.3.
Consider the situation of Definition 2.2 . Let $k \in N$ , $k ⩽ l$ and $1 ⩽ p < \infty$ . Then
$W^{k, p} (U, B)$ is a Banach space with norm $\begin{matrix} ‖ f ‖_{W^{k, p} (U, B)}^{} : = \sum_{j = 1}^{N} {‖ f \circ x_{j}^{- 1} |_{x_{j} (U \cap U_{j}) \cap W_{j}} ‖}_{W^{k, p} (x_{j} (U \cap U_{j}) \cap W_{j}, B)} for all f \in W^{k, p} (U, B) . \end{matrix}$ Different choices of* $(x_{j}, W_{j})$ yield the same spaces with equivalent norms.

$C^{l} (U, B) \cap W^{k, p} (U, B)$ is dense in $W^{k, p} (U, B)$ .

$\nabla_{U} f$ is well-defined for all $f \in W^{1, p} (U, B)$ and independent of the choices of $(x_{j}, W_{j})$ . Moreover, $\nabla_{U} f \in L^{p} (U, B^{n})$ and $\begin{matrix} ‖ f ‖_{W^{1, p} (U, B)} : = ‖ f ‖_{L^{p} (U, B)} + ‖ \nabla_{U} f ‖_{L^{p} (U, B^{n})} \end{matrix}$ defines an equivalent norm on $W^{1, p} (U, B)$ .

Proof.
The proof is technical but straightforward and uses density and localization arguments, cf. [66, Lemma 2.15]. □

Later on $W^{1, p} (U, B)$ we always take the norm in Lemma 2.3, 3. Note that for higher orders one can also define coordinate independent norms, cf. with Hebey [46, Chapter 2] in the scalar case. Nevertheless, later we only need $W^{1, p} (U, B)$ . For further properties of these spaces we refer to [66, Section 2.2.3].

Finally, we define the used spaces with exponential weight.
Definition 2.4.
Let $1 ⩽ p ⩽ \infty, k \in N_{0}, μ \in (0, 1)$ and $β, β_{1}, β_{2} ⩾ 0$ .
Then we introduce with canonical norms $\begin{array}{l} L_{(β_{1}, β_{2})}^{p} (R_{+}^{2}) : = {u = u (R, H) \in L_{loc}^{1} (R_{+}^{2}) : e^{β_{1} | R | + β_{2} H} u \in L^{p} (R_{+}^{2})}, \\ W_{(β_{1}, β_{2})}^{k, p} (R_{+}^{2}) : = {u \in L_{loc}^{1} (R_{+}^{2}) : D^{γ} u \in L_{(β_{1}, β_{2})}^{p} (R_{+}^{2}) for all | γ | ⩽ k} . \end{array}$ We also write $H^{k}$ instead of $W^{k, 2}$ . Moreover, $C_{(β_{1}, β_{2})}^{k} (\overline{R_{+}^{2}}) : = C_{b}^{k} (\overline{R_{+}^{2}}) \cap W_{(β_{1}, β_{2})}^{k, \infty} (R_{+}^{2})$ .

In a similar way we define $L_{(β)}^{p} (R)$ , $L_{(β)}^{p} (R_{+})$ , $W_{(β)}^{k, p} (R)$ , $W_{(β)}^{k, p} (R_{+})$ , $C_{(β)}^{k} (R)$ , $C_{(β)}^{k} (\overline{R_{+}})$ .

Let $\hat{η} : R \to R$ be smooth with $\hat{η} (R) = | R |$ for all $| R | ⩾ \overline{R}$ and some $\overline{R} > 0$ . Then we define $\begin{matrix} W_{(β)}^{k + μ, p} (R) : = {u = u (R) \in L_{loc}^{1} (R) : e^{β \hat{η} (R)} u \in W^{k + μ, p} (R)} \end{matrix}$ for $1 ⩽ p < \infty$ with natural norm.

The following lemma summarizes all the needed properties for these spaces: Lemma 2.5.

The spaces in Definition 2.4 are Banach spaces.

Equivalent norms: Let $\hat{η} : R \to R$ be as in Definition 2.4 , 3., $k \in N_{0}$ , $1 ⩽ p ⩽ \infty$ , $β_{1}, β_{2} ⩾ 0$ . Then $W_{(β_{1}, β_{2})}^{k, p} (R_{+}^{2}) = {u = u (R, H) \in L_{loc}^{1} (R_{+}^{2}) : e^{β_{1} \hat{η} (R) + β_{2} H} u \in W^{k, p} (R_{+}^{2})}$ and $\begin{array}{l} \sum_{γ \in N_{0}^{2}, | γ | ⩽ k} {‖ e^{β_{1} | R | + β_{2} H} D^{γ} u ‖}_{L^{p} (R_{+}^{2})}, \sum_{γ \in N_{0}^{2}, | γ | ⩽ k} {‖ e^{β_{1} \hat{η} (R) + β_{2} H} D^{γ} u ‖}_{L^{p} (R_{+}^{2})} and \\ {‖ e^{β_{1} \hat{η} (R) + β_{2} H} u ‖}_{W^{k, p} (R_{+}^{2})} \end{array}$ are equivalent uniformly in $u \in W_{(β_{1}, β_{2})}^{k, p} (R_{+}^{2})$ . For fixed $B > 0$ the constants in the equivalence estimates can be taken uniformly in $β_{1}, β_{2} \in [0, B]$ . Analogous assertions hold for $R$ instead of $R_{+}^{2}$ .

Density of smooth functions with compact support: For all $k \in N_{0}$ , $1 ⩽ p < \infty$ and $β_{1}, β_{2} ⩾ 0$ it holds: $C_{0}^{\infty} (\overline{R_{+}^{2}})$ is dense in $W_{(β_{1}, β_{2})}^{k, p} (R_{+}^{2})$ , $C_{0}^{\infty} (\overline{R_{+}})$ is dense in $W_{(β_{2})}^{k, p} (R_{+})$ and $C_{0}^{\infty} (R)$ is dense in $W_{(β_{1})}^{k, p} (R)$ .

Embeddings: It holds $W_{(β_{1}, β_{2})}^{k, p} (R_{+}^{2}) ↪ W_{(γ_{1}, γ_{2})}^{k, p} (R_{+}^{2})$ for all $k \in N_{0}$ , $1 ⩽ p ⩽ \infty$ and $0 ⩽ γ_{1} ⩽ β_{1}$ , $0 ⩽ γ_{2} ⩽ β_{2}$ , as well as $\begin{matrix} L_{(β_{1}, β_{2})}^{p} (R_{+}^{2}) ↪ L_{(β_{1} - ε, β_{2} - ε)}^{q} (R_{+}^{2}) for all min {β_{1}, β_{2}} > ε > 0, 1 ⩽ q ⩽ p . \end{matrix}$ Analogous embeddings hold for spaces on $R$ and $R_{+}$ .

Traces of weighted functions on $R_{+}^{2}$ : For all $k \in N$ , $1 ⩽ p < \infty$ and $β ⩾ 0$ the trace operator $\begin{matrix} tr : W_{(β, 0)}^{k, p} (R_{+}^{2}) \subset W^{k, p} (R_{+}^{2}) \to W_{(β)}^{k - \frac{1}{p}, p} (R) \end{matrix}$ is well-defined, bounded and there is a coretract operator $R_{β}$ (independent of $k, p$ ), i.e. $\begin{matrix} R_{β} \in L (W_{(β)}^{k - \frac{1}{p}, p} (R), W_{(β, 0)}^{k, p} (R_{+}^{2})) with tr \circ R_{β} = id . \end{matrix}$ Finally, all operator norms for fixed $k, p$ are bounded uniformly in $β ⩾ 0$ if we take the third norm in Lemma 2.5 , 2.

$L^{2}$ -Poincaré Inequality for weighted functions on $R_{+}$ : For $β > 0$ and all $u \in H_{(β)}^{1} (R_{+})$ it holds $‖ u ‖_{L_{(β)}^{2} (R_{+})} ⩽ \frac{1}{β} ‖ \partial_{H} u ‖_{L_{(β)}^{2} (R_{+})}$ .

Reverse Fundamental Theorem for weighted $L^{2}$ -functions on $R_{+}$ : For $β > 0$ and v in $L_{(β)}^{2} (R_{+})$ it holds $- \int_{\cdot}^{\infty} v d s = : w \in H_{(β)}^{1} (R_{+})$ with $\partial_{H} w = v$ . In particular 6. is applicable.

Remark 2.6.

Note that the choice $\int_{0}^{\cdot} v d s$ in Lemma 2.5, 7. would not be appropriate for integrability on $R_{+}$ .

From now on we will always use the third norm in Lemma 2.5, 2. for the weighted spaces.

Proof of Lemma 2.5.
The proofs of 1.–5. are straightforward and use the properties of the unweighted spaces. Cf. [66, Lemma 2.22].
Ad 6.
By density it is enough to prove the estimate for $u \in C_{0}^{\infty} (\overline{R_{+}})$ . With the Fundamental Theorem of Calculus, Fubini’s Theorem and the Hölder inequality we obtain $\begin{array}{l} ‖ u ‖_{L_{(β)}^{2} (R_{+})}^{2} & = \int_{0}^{\infty} e^{2 β H} u^{2} (H) d H ⩽ 2 \int_{0}^{\infty} e^{2 β H} \int_{H}^{\infty} | u \partial_{s} u | d s d H \\ ⩽ 2 \int_{0}^{\infty} \int_{0}^{s} e^{2 β H} d H | u \partial_{s} u | d s ⩽ \frac{1}{β} \int_{0}^{\infty} e^{2 β s} | u \partial_{s} u | d s ⩽ \frac{1}{β} ‖ \partial_{H} u ‖_{L_{(β)}^{2} (R_{+})} ‖ u ‖_{L_{(β)}^{2} (R_{+})}, \end{array}$ where we used $\int_{0}^{s} e^{2 β H} d H = \frac{1}{2 β} [e^{2 β s} - 1] ⩽ \frac{1}{2 β} e^{2 β s}$ . This shows the estimate. □
Ad 7.
Let $v \in L_{(β)}^{2} (R_{+})$ for a $β > 0$ and $v_{l} \in C_{0}^{\infty} (\overline{R_{+}})$ with $v_{l} \to v$ for $l \to \infty$ . Then $u_{l} : = - \int_{\cdot}^{\infty} v_{l} (s) d s \in C_{0}^{\infty} (\overline{R_{+}})$ with $\frac{d}{d H} u_{l} = v_{l}$ . From 6. we obtain that ${(u_{l})}_{l \in N}$ is a Cauchy sequence in $H_{(β)}^{1} (R_{+})$ , hence by 1. there is a limit u in $H_{(β)}^{1} (R_{+})$ and $\frac{d}{d H} u = v$ . Because of $u_{l} = - \int_{\cdot}^{\infty} v_{l} (s) d s \to - \int_{\cdot}^{\infty} v (s) d s$ for $l \to \infty$ pointwise, we get $u = - \int_{\cdot}^{\infty} v (s) d s$ . □

3. Curvilinear coordinates

Let $N ⩾ 2$ and $Ω \subseteq R^{N}$ be a bounded, smooth domain (i.e. nonempty, open and connected2

²
For convenience. The considerations can be adapted for the case of finitely many connected components.

) with outer unit normal

N_{\partial Ω}

. In this section we show the existence of a curvilinear coordinate system describing a neighbourhood of a suitable evolving surface3

For the definition of an evolving hypersurface cf. Depner [32, Definition 2.31].

\overline{Ω}

that meets the boundary

\partial Ω

at a 90°-contact angle. We adapt the ideas from the 2-dimensional case in [5] to the N-dimensional case.

3.1. Requirements for the evolving surface

Let $Σ \subset R^{N}$ , $N ⩾ 2$ be a smooth, orientable, compact and connected² hypersurface with boundary $\partial Σ$ and let $X_{0} : Σ \times [0, T] \to \overline{Ω}$ be smooth such that $X_{0} (\cdot, t)$ is an injective immersion for all $t \in [0, T]$ . For technical reasons, we assume that there is a smooth, orientable and connected hypersurface ${\tilde{Σ}}^{'} \subset R^{N}$ without boundary such that $Σ ⊊ {\tilde{Σ}}^{'}$ and a smooth extension of $X_{0}$ to ${\tilde{X}}_{0} : {\tilde{Σ}}^{'} \times (- \tilde{τ}, T + \tilde{τ}) \to R^{N}$ for some $\tilde{τ} > 0$ such that ${\tilde{X}}_{0} (\cdot, t)$ is an injective immersion for all $t \in (- \tilde{τ}, T + \tilde{τ})$ . Finally, we choose a smooth, orientable, compact and connected hypersurface $\tilde{Σ}$ with boundary such that $Σ ⊊ {\tilde{Σ}}^{\circ}$ and $\tilde{Σ} ⊊ {\tilde{Σ}}^{'}$ .

Remark 3.1.
Such ${\tilde{Σ}}^{'}, \tilde{τ}, {\tilde{X}}_{0}$ should exist for any Σ, $X_{0}$ as above. For $N = 2$ this is clear, but for $N ⩾ 3$ this is more difficult to show. First, it should be possible to extend any Σ as above to a smooth orientable hypersurface $\hat{Σ} \subset R^{N}$ without boundary by merging together local extensions in a suitable way. However, this is quite technical since one has to deal with fraying. Then $X_{0}$ can be extended to a smooth immersion $\hat{X}$ on an open neighbourhood of $Σ \times [0, T]$ in $\hat{Σ} \times R$ . Because immersions are locally injective (cf. O’Neill [69, Lemma 1.33]), one can prove injectivity of $\hat{X}$ on a possibly smaller open neighbourhood of $Σ \times [0, T]$ in $\hat{Σ} \times R$ with a contradiction argument and compactness.

Since continuous bijections of compact topological spaces into Hausdorff spaces are homeo-morphisms, we know that $X_{0} (\cdot, t)$ is an embedding and $Γ_{t} : = X_{0} (Σ, t) \subset R^{N}$ is a smooth, orientable, compact and connected hypersurface with boundary for all $t \in [0, T]$ . Moreover, $\begin{matrix} Γ : = ⋃_{t \in [0, T]} Γ_{t} \times {t} \end{matrix}$ is a smooth evolving hypersurface and $\begin{matrix} {\overline{X}}_{0} : = (X_{0}, {pr}_{t}) : Σ \times [0, T] \to Γ : (s, t) \mapsto (X_{0} (s, t), t) \end{matrix}$ is a homeomorphism. We choose a smooth normal field $\vec{n} : Σ \times [0, T] \to R^{N}$ meaning that $\vec{n}$ is smooth and $\vec{n} (\cdot, t)$ describes a normal field on $Γ_{t}$ . Due to Depner [32, Lemma 2.40] the corresponding normal velocity is given by $\begin{matrix} V (s, t) : = V_{Γ_{t}} (s) : = \vec{n} (s, t) \cdot \partial_{t} X_{0} (s, t) for (s, t) \in Σ \times [0, T] . \end{matrix}$ Moreover, let $H (s, t) : = H_{Γ_{t}} (s)$ for $(s, t) \in Σ \times [0, T]$ be the mean curvature which we choose to be the sum of the principal curvatures. The above definitions applied to ${\tilde{X}}_{0}$ on $\tilde{Σ} \times [- \frac{\tilde{τ}}{2}, T + \frac{\tilde{τ}}{2}]$ yield suitable extensions of $Γ_{t}, Γ, \vec{n}, V$ and H. For convenience, we use the same notation for $\vec{n}$ .

Additionally, we require ${(Γ_{t})}^{\circ} \subseteq Ω$ and $\partial Γ_{t} \subseteq \partial Ω$ . Then the contact angle of $Γ_{t}$ with $\partial Ω$ in any boundary point $X_{0} (s, t), (s, t) \in \partial Σ \times [0, T]$ with respect to $\vec{n} (s, t)$ is defined by $\begin{matrix} | ∡ (N_{\partial Ω} |_{X_{0} (s, t)}, \vec{n} (s, t)) | \in (0, π), \end{matrix}$ where $∡ (N_{\partial Ω} |_{X_{0} (s, t)}, \vec{n} (s, t))$ is taken in $(- π, π)$ . We assume constant contact angle $\frac{π}{2}$ for all times $t \in [0, T]$ .
3.2. Existence of curvilinear coordinates

Let the assumptions in Section 3.1 hold. We consider the outer unit conormal ${\vec{n}}_{\partial Σ} : \partial Σ \to R^{N}$ , cf. [32, Definition 2.28]. Moreover, we introduce the outer unit conormal for the evolving hypersurface Γ, ${\vec{n}}_{\partial Γ} : \partial Σ \times [0, T] \to R^{N}$ , where ${\vec{n}}_{\partial Γ} (σ, t) : = {\vec{n}}_{\partial Γ_{t}} (σ)$ is the outer unit conormal with respect to $\partial Γ_{t}$ at $X_{0} (σ, t)$ for all $(σ, t) \in \partial Σ \times [0, T]$ . One can show smoothness and $\begin{matrix} {\vec{n}}_{\partial Γ_{t}} (σ) = N_{\partial Ω} |_{X_{0} (σ, t)} for all (σ, t) \in \partial Σ \times [0, T] \end{matrix}$ with the considerations in [32]. Furthermore, we use the tubular neighbourhood coordinate system of $\partial Σ$ in $\tilde{Σ}$ : for $μ_{1} > 0$ small there is a smooth diffeomorphism $\begin{matrix} \tilde{Y} : \partial Σ \times [- 2 μ_{1}, 2 μ_{1}] \to R (\tilde{Y}) \subset \tilde{Σ}, (σ, b) \mapsto \tilde{Y} (σ, b) \end{matrix}$ onto the range $R (\tilde{Y})$ of $\tilde{Y}$ which is a neighbourhood of $\partial Σ$ in $\tilde{Σ}$ such that $\tilde{Y} |_{b = 0} = {id}_{\partial Σ}$ and $Y : = \tilde{Y} |_{\partial Σ \times [0, 2 μ_{1}]}$ is a diffeomorphism onto a neighbourhood $R (Y)$ of $\partial Σ$ in Σ. We use the notation $(\tilde{σ}, \tilde{b}) : = {\tilde{Y}}^{- 1}$ . We define $\tilde{Y}$ via the exponential map on the normal bundle of $\partial Σ$ in $\tilde{Σ}$ , cf. O’Neill [69, Proposition 7.26]. Then $\begin{matrix} (3.1) & \partial_{b} Y (σ, 0) = - {\vec{n}}_{\partial Σ} (σ) for all σ \in \partial Σ . \end{matrix}$

Theorem 3.2 (Curvilinear Coordinates).

Let the above assumptions in Section 3 up to this point hold. There exist $δ > 0$ and a smooth map $[- δ, δ] \times Σ \times [0, T] ∋ (r, s, t) \mapsto X (r, s, t) \in \overline{Ω}$ with the following properties:

$\overline{X} : = (X, {pr}_{t})$ is a homeomorphism onto a neighbourhood of Γ in $\overline{Ω} \times [0, T]$ . Moreover, $\overline{X}$ can be extended to a smooth diffeomorphism defined on an open neighbourhood of $[- δ, δ] \times Σ \times [0, T]$ in $R \times \tilde{Σ} \times R$ mapping onto an open set in $R^{N + 1}$ . The set $\begin{matrix} Γ (\tilde{δ}) : = \overline{X} ((- \tilde{δ}, \tilde{δ}) \times Σ \times [0, T]) \end{matrix}$ is an open neighbourhood of Γ in $\overline{Ω} \times [0, T]$ for $\tilde{δ} \in (0, δ]$ .

$X |_{r = 0} = X_{0}$ and X coincides with the well-known tubular neighbourhood coordinate system for $s \in Σ ∖ Y (\partial Σ \times [0, μ_{0}])$ for some $μ_{0} \in (0, μ_{1}]$ small. Additionally, for points $(r, s, t) \in [- δ, δ] \times Σ \times [0, T]$ it holds $X (r, s, t) \in \partial Ω$ if and only if $s \in \partial Σ$ .

Let $(r, s, {pr}_{t})$ be the inverse of $\overline{X}$ . Then ${(\partial_{x_{j}} s |_{(x, t)})}_{j = 1}^{N}$ generate the tangent space $T_{s (x, t)} Σ$ , $| \nabla r |_{(x, t)} | ⩾ c > 0$ for some $c > 0$ independent of $(x, t)$ and $D_{x} s {(D_{x} s)}^{⊤} |_{(x, t)}$ is uniformly positive definite as a linear map in $L (T_{s (x, t)} Σ)$ for all $(x, t) \in \overline{Γ (δ)}$ . Furthermore, we have $\begin{matrix} | \nabla r |^{2} |_{Γ} = 1, \partial_{r} (| \nabla r |^{2} \circ \overline{X}) |_{r = 0} = 0 and D_{x} s \nabla r |_{Γ} = 0 \end{matrix}$ and for all $(r, s, t) \in [- δ, δ] \times [Σ ∖ Y (\partial Σ \times [0, μ_{0}])] \times [0, T]$ it holds $\begin{matrix} \nabla r |_{\overline{X} (r, s, t)} = \vec{n} (s, t) and D_{x} s |_{\overline{X} (r, s, t)} \vec{n} (s, t) = 0 . \end{matrix}$ Moreover, we can choose $\nabla r \circ {\overline{X}}_{0} = \vec{n}$ . Then it holds $V = - \partial_{t} r \circ {\overline{X}}_{0}$ and $H = - Δ r \circ {\overline{X}}_{0}$ .

Let $(σ, b) : = Y^{- 1} \circ s : \overline{X} ([- δ, δ] \times R (Y) \times [0, T]) \to \partial Σ \times [0, 2 μ_{1}]$ . Then $\begin{array}{l} N_{\partial Ω} \cdot \nabla b |_{{\overline{X}}_{0} (σ, t)} = - D_{x} s N_{\partial Ω} |_{{\overline{X}}_{0} (σ, t)} \cdot {\vec{n}}_{\partial Σ} |_{σ}, | N_{\partial Ω} \cdot \nabla b |_{{\overline{X}}_{0} (σ, t)} | ⩾ c > 0 \end{array}$ and $\nabla b \cdot \nabla r |_{{\overline{X}}_{0} (σ, t)} = 0$ , $| \nabla b |_{{\overline{X}}_{0} (σ, t)} | ⩾ c > 0$ for all $(σ, t) \in \partial Σ \times [0, T]$ .

Remark 3.3.

Let $Q_{T} : = Ω \times (0, T)$ . There are unique connected $Q_{T}^{\pm} \subseteq \overline{Q_{T}} = \overline{Ω} \times [0, T]$ such that $\overline{Q_{T}} = Q_{T}^{-} \cup Q_{T}^{+} \cup Γ$ (disjoint) and $sign r = \pm 1$ on $Q_{T}^{\pm} \cap Γ (δ)$ . Moreover, we set $\begin{matrix} Γ^{C} (\tilde{δ}, μ) : = \overline{X} ((- \tilde{δ}, \tilde{δ}) \times Y (\partial Σ \times (0, μ)) \times [0, T]), Γ (\tilde{δ}, μ) : = Γ (\tilde{δ}) ∖ \overline{Γ^{C} (\tilde{δ}, μ)} \end{matrix}$ for $\tilde{δ} \in (0, δ]$ and $μ \in (0, 2 μ_{1}]$ . For $t \in [0, T]$ fixed let $Γ_{t} (\tilde{δ}), Γ_{t}^{C} (\tilde{δ}, μ)$ and $Γ_{t} (\tilde{δ}, μ)$ be the respective sets intersected with $R^{N} \times {t}$ and then projected to $R^{N}$ . Here $Γ (\tilde{δ})$ is defined in Theorem 3.2.

Let $\tilde{δ} \in (0, δ]$ . For a sufficiently smooth $ψ : Γ (\tilde{δ}) \to R$ we define the tangential and normal derivative by $\begin{matrix} \nabla_{τ} ψ : = {(D_{x} s)}^{⊤} [\nabla_{Σ} (ψ \circ \overline{X}) \circ {\overline{X}}^{- 1}] and \partial_{n} ψ : = \partial_{r} (ψ \circ \overline{X}) \circ {\overline{X}}^{- 1}, \end{matrix}$ respectively. For $t \in [0, T]$ fixed and $ψ : Γ_{t} (\tilde{δ}) \to R$ smooth enough, we define $\nabla_{τ} ψ$ and $\partial_{n} ψ$ analogously. The same notation applies if ψ is only defined on an open subset of $Γ (\tilde{δ})$ or $Γ_{t} (\tilde{δ})$ for some $μ \in (0, 2 μ_{1}]$ and $t \in [0, T]$ . Note that in the case $N = 2$ and $Σ = [- 1, 1]$ the definitions here coincide with the ones in [5, Remark 2.2]. For suitable $R^{m}$ -valued functions $\vec{ψ}$ we define $\partial_{n} \vec{ψ}$ and $\nabla_{τ} \vec{ψ}$ component-wise. More precisely $\partial_{n} \vec{ψ} : = (\partial_{n} ψ_{1}, \dots, \partial_{n} ψ_{m})$ and $\nabla_{τ} \vec{ψ} : = ({(\nabla_{τ} ψ_{1})}^{⊤}, \dots, {(\nabla_{τ} ψ_{m})}^{⊤})$ . Important properties of $\nabla_{τ}$ and $\partial_{n}$ will be shown in Corollary 3.5.

For transformation arguments we define $\begin{matrix} J (r, s, t) : = J_{t} (r, s) : = | det d_{(r, s)} X (r, s, t) | for (r, s, t) \in [- δ, δ] \times Σ \times [0, T], \end{matrix}$ where the determinant is taken with respect to an arbitrary orthonormal basis of $T_{s} Σ$ . Via local coordinates it follows that J is smooth and with a compactness argument we obtain that $0 < c ⩽ J ⩽ C$ for some $c, C > 0$ .

The proof of Theorem 3.2 is similar to [5, Theorem 2.1], where the case $N = 2$ was proven. The first step for the proof of Theorem 3.2 is to show an analogue of [5, Lemma 2.3].
Lemma 3.4.
There is an $η > 0$ such that $\partial Ω \cap R_{η} (σ, t)$ admits a graph parametrization over $X_{0} (σ, t) + [B_{η} (0) \cap T_{X_{0} (σ, t)} \partial Ω]$ for all $(σ, t) \in \partial Σ \times [0, T]$ , where $\begin{matrix} R_{η} (σ, t) : = X_{0} (σ, t) + (- η, η) {\vec{n}}_{\partial Γ} (σ, t) + [B_{η} (0) \cap T_{X_{0} (σ, t)} \partial Ω] . \end{matrix}$ Moreover, for $η > 0$ small there exists $w : (- η, η) \times \partial Σ \times [0, T] \to R$ smooth such that $w |_{r = 0} = \partial_{r} w |_{r = 0} = 0$ and $\begin{matrix} (- η, η) ∋ r \mapsto X_{0} (σ, t) + r \vec{n} (σ, t) + w (r, σ, t) {\vec{n}}_{\partial Γ} (σ, t) \end{matrix}$ describes $\partial Ω$ in $X_{0} (σ, t) + (- η, η) \vec{n} (σ, t) + (- η, η) {\vec{n}}_{\partial Γ} (σ, t)$ for all $(σ, t) \in \partial Σ \times [0, T]$ .

The assertions are compatible with shrinking η for small $η > 0$ which follows from the contact angle assumption and Taylor’s Theorem.
Proof.
Let $(σ_{0}, t_{0}) \in \partial Σ \times [0, T]$ be arbitrary. We choose a basis ${\vec{v}}_{1}, \dots, {\vec{v}}_{N - 2}$ of $T_{X_{0} (σ_{0}, t_{0})} \partial Γ_{t_{0}}$ and extend it to smooth tangential vector fields ${\vec{τ}}_{1}, \dots, {\vec{τ}}_{N - 2}$ on $T \partial Γ$ such that locally in $\partial Γ$ around $X_{0} (σ_{0}, t_{0})$ these are again bases in the corresponding tangential spaces over $\partial Γ_{t}$ for all $t \in B_{ε} (t_{0}) \cap [0, T]$ , $ε > 0$ small. In coordinates one can apply similar arguments as in the proof of [5, Lemma 2.3], to obtain smooth graph parametrizations of $\partial Ω \cap R_{η} (σ, t)$ over $X_{0} (σ, t) + [B_{η} (0) \cap T_{X_{0} (σ, t)} \partial Ω]$ for some $η > 0$ and $(σ, t) \in \partial Σ \times [0, T]$ close to $(σ_{0}, t_{0})$ . More precisely, there is an $η > 0$ and a smooth $\begin{matrix} w_{0} : (- η, η) \times B_{η} (0) \times U \times V \subseteq R \times R^{N - 2} \times Σ \times [0, T] \to R, \end{matrix}$ where $U, V$ are open neighbourhoods of $σ_{0}, t_{0}$ in $\partial Σ, [0, T]$ , respectively, such that $\begin{array}{l} (- η, η) \times B_{η} (0) ∋ (r, r_{1}, \dots, r_{N - 2}) \mapsto & X_{0} (σ, t) + r \vec{n} (σ, t) + r_{1} {\vec{τ}}_{1} (σ, t) + \dots + r_{N - 2} {\vec{τ}}_{N - 2} (σ, t) \\ + w_{0} (r, r_{2}, \dots, r_{N - 2}, σ, t) {\vec{n}}_{\partial Γ} (σ, t) \end{array}$ describes $\partial Ω \cap R_{η} (σ, t)$ in $R_{η} (σ, t)$ . Moreover, $w_{0} |_{r = 0} = \partial_{r} w_{0} |_{r = 0} = 0$ . We set $\begin{matrix} w : (- η, η) \times U \times V \to R : (r, σ, t) \mapsto w_{0} (r, 0, \dots, 0, σ, t) \end{matrix}$ and we observe that this definition is independent of the choice of ${\vec{v}}_{1}, \dots, {\vec{v}}_{N - 2}$ and ${\vec{τ}}_{1}, \dots, {\vec{τ}}_{N - 2}$ as well as $(σ_{0}, t_{0})$ . By compactness $η > 0$ can be taken uniformly in $(σ_{0}, t_{0}) \in \partial Σ \times [0, T]$ . □
Proof of Theorem 3.2.
Let $Σ_{1}$ be a compact hypersurface with boundary such that $Σ ⊊ Σ_{1}^{\circ}$ and $Σ_{1} ⊊ {\tilde{Σ}}^{\circ}$ . Similar as in the proof of [5, Theorem 2.1] there is a $δ_{0} > 0$ such that for all $δ \in (0, δ_{0}]$ it holds that $\begin{matrix} (- δ, δ) \times Σ_{1} ∋ (r, s) \mapsto {\tilde{X}}_{0} (s, t) + r \vec{n} (s, t) \in R^{N} \end{matrix}$ is a diffeomorphism onto its image $U_{δ} (t)$ and $U_{δ} (t) \cap \overline{Ω} = B_{δ} ({\tilde{Γ}}_{t}) \cap \overline{Ω}$ for all $t \in [0, T]$ , where we have set ${\tilde{Γ}}_{t} : = {\tilde{X}}_{0} (\tilde{Σ}, t)$ .

We choose $η > 0$ small such that $R_{η} (\partial Σ, t)$ is contained in $U_{δ_{0}} (t)$ , the assertions of Lemma 3.4 are fulfilled and such that the angles between the tangent planes of $R_{η} (σ, t) \cap \partial Ω$ are smaller than a fixed $β > 0$ (which will be chosen later).

Now we define X. Let $\vec{τ} : \tilde{Σ} \times [0, T] \to R^{N}$ be a smooth vector field with the property that $\vec{τ} (s, t) \in T_{{\tilde{X}}_{0} (s, t)} {\tilde{Γ}}_{t}$ for all $(s, t) \in \tilde{Σ} \times [0, T]$ and $\vec{τ} |_{\partial Σ \times [0, T]} = {\vec{n}}_{\partial Γ}$ . Existence of such a $\vec{τ}$ follows via local extensions of ${\vec{n}}_{\partial Γ}$ in submanifold charts of $\partial Σ$ with respect to $\tilde{Σ}$ and compactness arguments. Moreover, by uniform continuity there is an $ε \in (0, μ_{1}]$ such that ${\tilde{X}}_{0} (\tilde{Y} (σ, \tilde{b}), t) \in R_{\frac{η}{2}} (σ, t)$ for all $(σ, \tilde{b}, t) \in \partial Σ \times [- ε, ε] \times [0, T]$ as well as $\begin{matrix} (3.2) & | \partial_{b} \tilde{Y} (σ, \tilde{b}) + {\vec{n}}_{\partial Σ} (σ) | + | \frac{d}{d \tilde{b}} {\tilde{X}}_{0} (\tilde{Y} (σ, \tilde{b})) - \frac{d}{d \tilde{b}} |_{\tilde{b} = 0} {\tilde{X}}_{0} (\tilde{Y} (σ, \cdot)) | ⩽ c_{0} \end{matrix}$ for a fixed $c_{0} > 0$ small (to be determined later), where we used (3.1). Let $χ : R \to [0, 1]$ be a smooth cutoff-function with $χ = 1$ for $| b | ⩽ \frac{ε}{2}$ and $χ = 0$ for $| b | ⩾ ε$ . We define $\vec{T} (s, t) : = χ (\tilde{b} (s)) \vec{τ} (s, t)$ for $(s, t) \in Σ \times [0, T]$ and $\begin{matrix} X (r, s, t) : = {\tilde{X}}_{0} (s, t) + r \vec{n} (s, t) + w (r, \tilde{σ} (s), t) \vec{T} (s, t) \in R^{N} for (r, s, t) \in [- δ, δ] \times \tilde{Σ} \times [0, T] \end{matrix}$ and $δ > 0$ small. In the following we show that the properties in the theorem are satisfied if $δ > 0$ is small and $β > 0$ as well as $c_{0} > 0$ above were chosen properly.
Ad 1.–2.
First of all, X is well-defined due to the cutoff-function. Moreover, X is smooth and $\begin{array}{l} (3.3) & \partial_{r} X (r, s, t) = \vec{n} (s, t) + \partial_{r} w (r, \tilde{σ} (s), t) \vec{T} (s, t) \in R^{N}, \\ (3.4) & d_{s} [X (r, \cdot, t)] = d_{s} [{\tilde{X}}_{0} (\cdot, t)] + r d_{s} [\vec{n} (\cdot, t)] + d_{s} [w (r, \tilde{σ} (\cdot), t)] \vec{T} + w |_{(r, \tilde{σ} (s), t)} d_{s} [\vec{T} (\cdot, t)] \end{array}$ for all $(r, s, t) \in [- δ, δ] \times \tilde{Σ} \times [0, T]$ , where $d_{s} [X (r, \cdot, t)]$ , $d_{s} [{\tilde{X}}_{0} (\cdot, t)]$ , $d_{s} [\vec{n} (\cdot, t)]$ and $d_{s} [\vec{T} (\cdot, t)]$ map from $T_{s} \tilde{Σ}$ to $R^{N}$ . Hence $\begin{array}{l} (3.5) & d_{(r, s)} [X (\cdot, t)] : R \times T_{s} \tilde{Σ} \to R^{N}, (v_{1}, v_{2}) \mapsto \partial_{r} X |_{(r, s, t)} v_{1} + d_{s} [X (r, \cdot, t)] (v_{2}), \\ (3.6) & d_{(0, s)} [X (\cdot, t)] : R \times T_{s} \tilde{Σ} \to R^{N}, (v_{1}, v_{2}) \mapsto \vec{n} (s, t) v_{1} + d_{s} [{\tilde{X}}_{0} (\cdot, t)] (v_{2}), \end{array}$ where we used $w |_{r = 0} = \partial_{r} w |_{r = 0} = 0$ . Since $d_{s} [{\tilde{X}}_{0} (\cdot, t)] : T_{s} \tilde{Σ} \to T_{{\tilde{X}}_{0} (s, t)} {\tilde{Γ}}_{t}$ is an isomorphism and $R^{N} = T_{{\tilde{X}}_{0} (s, t)} {\tilde{Γ}}_{t} \oplus N_{{\tilde{X}}_{0} (s, t)} {\tilde{Γ}}_{t}$ , we obtain that $d_{(0, s)} [X (\cdot, t)] : R \times T_{s} \tilde{Σ} \to R^{N}$ is invertible for every $(s, t) \in \tilde{Σ} \times [0, T]$ . By compactness this is also valid for $d_{(r, s)} [X (\cdot, t)]$ for every $(r, s, t) \in [- δ, δ] \times \tilde{Σ} \times [0, T]$ if $δ > 0$ is small, cf. the similar argument in the proof of [5, Theorem 2.1]. The Inverse Function Theorem yields that $\overline{X}$ is locally injective and together with injectivity on ${0} \times \tilde{Σ} \times [0, T]$ we get similarly as in the 2-dimensional case in [5] by contradiction and compactness that $\overline{X}$ is injective on $[- δ, δ] \times \tilde{Σ} \times [0, T]$ for $δ > 0$ small. Moreover, due to the Inverse Function Theorem, $\overline{X}$ can locally in $R \times {\tilde{Σ}}^{'} \times R$ be extended to a smooth diffeomorphism. With a similar contradiction and compactness argument as before, it follows that $\overline{X}$ can be extended to a smooth diffeomorphism on an open neighbourhood of $[- δ, δ] \times Σ \times [0, T]$ in $R \times Σ_{0} \times R$ mapping onto an open set in $R^{N + 1}$ .

Next we prove that $X ([- δ, δ] \times Σ \times [0, T]) \subset \overline{Ω}$ if $δ > 0$ is small and related properties. First, note that the set $Γ ∖ {\overline{X}}_{0} (Y (\partial Σ \times [0, ε] \times [0, T]))$ has a positive distance to $\partial Ω \times [0, T]$ by compactness. Moreover, $\begin{matrix} X (r, s, t) = X_{0} (s, t) + r \vec{n} (s, t) for (r, s, t) \in [- δ, δ] \times [Σ ∖ Y (\partial Σ \times [0, ε])] \times [0, T] . \end{matrix}$ Therefore $X (r, s, t)$ stays in Ω for such $(r, s, t)$ if $δ > 0$ is small. For the remaining points we use geometric arguments with angles and Lemma 3.4. For $s \in Y (\partial Σ \times [0, ε])$ we observe that $Y (\tilde{σ} (s), \cdot) : [0, \tilde{b} (s)] \to Σ$ is a curve from $\tilde{σ} (s)$ to s. Hence $X (r, Y (\tilde{σ} (s), \cdot), t) : [0, b (s)] \to R^{N}$ is a curve from $X (r, \tilde{σ} (s), t)$ to $X (r, s, t)$ , where $\begin{matrix} \frac{d}{d b} [X (r, Y (\tilde{σ} (s), b), t)] = d_{Y (\tilde{σ} (s), b)} [X (r, \cdot, t)] (\partial_{b} Y (\tilde{σ} (s), b)) . \end{matrix}$ Because of (3.4) and $w |_{r = 0} = \partial_{r} w |_{r = 0} = 0$ we have $\begin{matrix} \frac{d}{d b} |_{b = 0} [X (0, Y (σ, \cdot), t)] = d_{σ} [X_{0} (\cdot, t)] (- {\vec{n}}_{\partial Σ} (σ)) for all (σ, t) \in \partial Σ \times [0, T] . \end{matrix}$ Here $d_{σ} [X_{0} (\cdot, t)]$ is invertible from $T_{σ} Σ$ to $T_{X_{0} (σ, t)} Γ_{t}$ as well as from $T_{σ} \partial Σ$ to $T_{X_{0} (σ, t)} \partial Γ_{t}$ . Therefore $d_{σ} [X_{0} (\cdot, t)] ({\vec{n}}_{\partial Σ} (σ)) \cdot {\vec{n}}_{\partial Γ} (σ, t) > 0$ and by compactness $\begin{matrix} d_{σ} [X_{0} (\cdot, t)] ({\vec{n}}_{\partial Σ} (σ)) \cdot {\vec{n}}_{\partial Γ} (σ, t) ⩾ c_{1} > 0 for all (σ, t) \in \partial Σ \times [0, T] . \end{matrix}$ Due to (3.2) and (3.4) we obtain for all $(r, b, s, t) \in [- δ, δ] \times [0, ε] \times Y (\partial Σ \times [0, ε]) \times [0, T]$ that $\begin{matrix} (3.7) & - {\vec{n}}_{\partial Γ} |_{(\tilde{σ} (s), t)} \cdot \frac{d}{d b} [X (r, Y (\tilde{σ} (s), b), t)] ⩾ \frac{c_{1}}{2} > 0 \end{matrix}$ provided that $δ > 0$ is small and $c_{0} > 0$ was chosen sufficiently small before. Moreover, it holds $X_{0} (Y (σ, b), t) \in R_{\frac{η}{2}} (σ, t)$ for $(σ, b, t) \in \partial Σ \times [0, ε] \times [0, T]$ by the choice of ε. This yields $\begin{matrix} (3.8) & X (r, Y (σ, b), t) \in R_{\frac{3 η}{4}} (σ, t) for all (r, σ, b, t) \in [- δ, δ] \times \partial Σ \times [0, ε] \times [0, T] \end{matrix}$ if $δ > 0$ is small. Altogether we can determine the location of $X (r, Y (σ, b), t)$ geometrically: By (3.8) we know that $X (r, Y (σ, b), t)$ is contained in a cylinder where we have a suitable graph parametrization of $\partial Ω$ due to Lemma 3.4. Moreover, (3.7) and the Fundamental Theorem of Calculus yield that $X (r, Y (σ, b), t)$ lies in a cone (where $c_{1}$ determines how close it can be to a half space) viewed from $X (r, Y (σ, 0), t)$ . Therefore if $β > 0$ in the beginning of the proof was chosen sufficiently small, the cone without the tip lies inside of Ω. Note that $c_{1}$ above is independent of $β, c_{0}, η, ε, δ$ . Therefore we can choose $β, c_{0} > 0$ small first (both only depending on $c_{1}$ ), then $η > 0$ , then $ε > 0$ and finally $δ > 0$ . This proves $X ([- δ, δ] \times Σ \times [0, T]) \subset \overline{Ω}$ and $X (r, s, t) \in \partial Ω$ if and only if $s \in \partial Σ$ . Moreover, with an analogous argument it follows that X maps $\begin{matrix} [- δ, δ] \times \tilde{Y} (\partial Σ \times [- ε, 0)) \times [0, T] \end{matrix}$ outside $\overline{Ω}$ for possibly smaller $β, c_{0}, η, ε, δ$ . Finally, the extension property of $\overline{X}$ yields that $Γ (\tilde{δ})$ is an open neighbourhood of Γ in $\overline{Ω} \times [0, T]$ for $\tilde{δ} \in (0, δ]$ if $δ > 0$ is small. □
Ad 3.
Consider $(r, s, {pr}_{t}) : = {\overline{X}}^{- 1} : \overline{Γ (δ)} \to [- δ, δ] \times Σ \times [0, T]$ . Then $\begin{matrix} d_{x} [X^{- 1} (\cdot, t)] : R^{N} \to R \times T_{s (x, t)} Σ : \vec{v} \mapsto (d_{x} [r (\cdot, t)], d_{x} [s (\cdot, t)]) \vec{v} = (\nabla r |_{(x, t)} \cdot \vec{v}, D_{x} s |_{(x, t)} \vec{v}) \end{matrix}$ is invertible for all $(x, t) \in \overline{Γ (δ)}$ . Therefore $| \nabla r |_{(x, t)} | > 0$ and by compactness $| \nabla r |_{(x, t)} | ⩾ c > 0$ for all $(x, t) \in \overline{Γ (δ)}$ . Moreover, ${(\partial_{x_{j}} s |_{(x, t)})}_{j = 1}^{N}$ generate $T_{s (x, t)} Σ$ for all such $(x, t)$ . In particular $\begin{matrix} D_{x} s {(D_{x} s)}^{⊤} |_{(x, t)} = {(\nabla s_{i} \cdot \nabla s_{j} |_{(x, t)})}_{i, j = 1}^{N} = \sum_{q = 1}^{N} \partial_{q} s {(\partial_{q} s)}^{⊤} |_{(x, t)} \end{matrix}$ is injective as a linear map in $L (T_{s (x, t)} Σ)$ for all $(x, t) \in \overline{Γ (δ)}$ . The latter follows directly since $D_{x} s {(D_{x} s)}^{⊤} |_{(x, t)} \vec{v} = 0$ for some $\vec{v} \in T_{s (x, t)} Σ$ implies $\sum_{q = 1}^{N} | {(\partial_{q} s |_{(x, t)})}^{⊤} \vec{v} |^{2} = 0$ and hence $\vec{v} = 0$ . Therefore $D_{x} s {(D_{x} s)}^{⊤} |_{(x, t)}$ is positive definite on $T_{s (x, t)} Σ$ for all $(x, t) \in \overline{Γ (δ)}$ . In local coordinates the latter transforms to a linear map on $R^{N - 1}$ . Note that by scaling it is equivalent to consider vectors in the sphere in $R^{N - 1}$ in order to verify the definition of positive definiteness. Therefore by compactness we obtain that $D_{x} s {(D_{x} s)}^{⊤} |_{(x, t)}$ is uniformly positive definite as a linear map in $L (T_{s (x, t)} Σ)$ for all $(x, t) \in \overline{Γ (δ)}$ .

Let $(r, s, t) \in [- δ, δ] \times Σ \times [0, T]$ . Then $d_{(r, s)} [X (\cdot, t)] \circ d_{X (r, s, t)} [X^{- 1} (\cdot, t)] = {Id}_{R^{N}}$ . Hence $\begin{matrix} {(d_{(r, s)} [X (\cdot, t)])}^{- 1} = (d_{X (r, s, t)} [r (\cdot, t)], d_{X (r, s, t)} [s (\cdot, t)]) : R^{N} \to R \times T_{s} Σ . \end{matrix}$ On the other hand (3.6) implies $\begin{matrix} (3.9) & d_{X (0, s, t)} [r (\cdot, t)] = \vec{n} {(s, t)}^{⊤} and d_{X (0, s, t)} [s (\cdot, t)] = d_{s} {[X_{0} (\cdot, t)]}^{- 1} P_{T_{X_{0} (s, t)} Γ_{t}} . \end{matrix}$ We can also write $\begin{matrix} (3.10) & d_{X (r, s, t)} [r (\cdot, t)] (\vec{v}) = D_{x} r |_{\overline{X} (r, s, t)} \vec{v} and d_{X (r, s, t)} [s (\cdot, t)] (\vec{v}) = D_{x} s |_{\overline{X} (r, s, t)} \vec{v} \end{matrix}$ for all $\vec{v} \in R^{N}$ . Altogether this yields $\nabla r |_{{\overline{X}}_{0} (s, t)} = \vec{n} (s, t)$ and $D_{x} s |_{{\overline{X}}_{0} (s, t)} \vec{n} (s, t) = 0$ for all $(s, t) \in Σ \times [0, T]$ . With similar arguments we obtain $\begin{matrix} \nabla r |_{\overline{X} (r, s, t)} = \vec{n} (s, t) and D_{x} s |_{\overline{X} (r, s, t)} \vec{n} (s, t) = 0 \end{matrix}$ for all $(r, s, t) \in [- δ, δ] \times [Σ ∖ Y (\partial Σ \times [0, μ_{0}])] \times [0, T]$ . Moreover, it holds $\begin{matrix} \partial_{r} (| \nabla r |^{2} \circ \overline{X}) |_{(0, s, t)} = 2 \frac{d}{d r} [D_{x} r |_{X (\cdot, s, t)}] |_{r = 0} \cdot \nabla r |_{{\overline{X}}_{0} (s, t)} . \end{matrix}$ In order to use (3.10) we compute $\begin{matrix} \frac{d}{d r} [{(d_{(r, s)} [X (\cdot, t)])}^{- 1}] |_{r = 0} = - {(d_{(0, s)} X (\cdot, t))}^{- 1} \circ \frac{d}{d r} [d_{(r, s)} X (\cdot, t)] |_{r = 0} \circ {(d_{(0, s)} X (\cdot, t))}^{- 1} \end{matrix}$ with the formula for the Fréchet derivative of the inverse of a differentiable family of invertible, linear operators. Here ${(d_{(0, s)} X (\cdot, t))}^{- 1} : R^{N} \to R \times T_{s} Σ$ is explicitly determined by (3.9). Furthermore, differentiating (3.3)–(3.4) with respect to r we obtain for all $(v_{1}, v_{2}) \in R \times T_{s} Σ$ $\begin{matrix} \frac{d}{d r} [d_{(r, s)} X (\cdot, t)] |_{r = 0} (v_{1}, v_{2}) = \partial_{r}^{2} w (0, σ (s), t) \vec{T} (s, t) v_{1} + d_{s} [\vec{n} (\cdot, t)] (v_{2}) \in R^{N} . \end{matrix}$ Therefore it holds $\begin{array}{l} \frac{d}{d r} [{(d_{(r, s)} [X (\cdot, t)])}^{- 1}] |_{r = 0} (\vec{n} (s, t)) & = - {(d_{(0, s)} X (\cdot, t))}^{- 1} (\frac{d}{d r} [d_{(r, s)} X (\cdot, t)] |_{r = 0} (1, 0)) \\ = - {(d_{(0, s)} X (\cdot, t))}^{- 1} (\partial_{r}^{2} w (0, σ (s), t) \vec{T} (s, t)) \\ = - (0, \partial_{r}^{2} w (0, σ (s), t) d_{s} {[X_{0} (\cdot, t)]}^{- 1} \vec{T} (s, t)), \end{array}$ where $\partial_{r} (| \nabla r |^{2} \circ \overline{X}) |_{(0, s, t)}$ equals twice the first component, thus equals zero. Moreover, one can prove the identities for the normal velocity V and mean curvature H in an analogous way as in the case $N = 2$ , cf. the proof of [5, Theorem 2.1]. □
Ad 4.
Finally, we show the properties of b. Let $(\overline{σ}, \overline{b}) : = Y^{- 1} : R (Y) \to \partial Σ \times [0, 2 μ_{1}]$ . Then $b = \overline{b} \circ s$ on $\overline{X} ([- δ, δ] \times R (Y) \times [0, T])$ and by chain rule $\begin{matrix} d_{x} [b (\cdot, t)] = d_{s (x, t)} \overline{b} \circ d_{x} [s (\cdot, t)] : R^{N} \to R, \end{matrix}$ where we are interested in boundary points $x = X_{0} (σ, t)$ for any $(σ, t) \in \partial Σ \times [0, T]$ . For such x it holds $s (x, t) = σ$ . Because of $Y (\cdot, 0) = {id}_{\partial Σ}$ and (3.1) we have $\begin{matrix} d_{(σ, 0)} Y : T_{σ} \partial Σ \times R \to T_{σ} Σ : (v_{1}, v_{2}) \mapsto v_{1} - {\vec{n}}_{\partial Σ} (σ) v_{2} . \end{matrix}$ Therefore $(d_{σ} \overline{σ}, d_{σ} \overline{b}) = {(d_{(σ, 0)} Y)}^{- 1} = ({pr}_{T_{σ} \partial Σ}, - {\vec{n}}_{\partial Σ} {(σ)}^{⊤})$ . Together with (3.9) we obtain $\begin{matrix} \nabla b |_{(x, t)} \cdot v = d_{x} [b (\cdot, t)] (v) = - {\vec{n}}_{\partial Σ} (σ) \cdot ({(d_{σ} [X_{0} (\cdot, t)])}^{- 1} \circ P_{T_{x} Γ_{t}} (v)) for all v \in R^{N} . \end{matrix}$ Hence $\nabla b |_{{\overline{X}}_{0} (σ, t)} = \nabla b |_{(x, t)} \in T_{X_{0} (σ, t)} Γ_{t}$ , in particular $\nabla b \cdot \nabla r |_{{\overline{X}}_{0} (σ, t)} = 0$ , and $\begin{matrix} \nabla b |_{{\overline{X}}_{0} (σ, b)} \cdot N_{\partial Ω} |_{X_{0} (σ, b)} = - {\vec{n}}_{\partial Σ} (σ) \cdot {(d_{σ} [X_{0} (\cdot, t)])}^{- 1} {\vec{n}}_{\partial Γ} (σ, t) for all (σ, t) \in \partial Σ \times [0, T] . \end{matrix}$ Note that due to (3.9)–(3.10) it holds $\begin{matrix} {(d_{σ} [X_{0} (\cdot, t)])}^{- 1} {\vec{n}}_{\partial Γ} (σ, t) = D_{x} s N_{\partial Ω} |_{{\overline{X}}_{0} (σ, t)} \end{matrix}$ for all $(σ, t) \in \partial Σ \times [0, T]$ . Hence we obtain the identity in the theorem. Moreover, we know that $d_{σ} [X_{0} (\cdot, t)]$ is invertible from $T_{σ} Σ$ to $T_{X_{0} (σ, t)} Γ_{t}$ as well as from $T_{σ} \partial Σ$ to $T_{X_{0} (σ, t)} \partial Γ_{t}$ . This yields that $| \nabla b |_{{\overline{X}}_{0} (σ, t)} \cdot N_{\partial Ω} |_{X_{0} (σ, t)} | > 0$ for all $(σ, t) \in \partial Σ \times [0, T]$ and by smoothness and compactness the latter is bounded from below by a uniform positive constant. Because of the Cauchy–Schwarz-Inequality, this estimate carries over to $| \nabla b |_{{\overline{X}}_{0} (σ, t)} |$ . □

Finally, we show relations of $\partial_{n}, \nabla_{τ}$ defined in Remark 3.3, 2. to $\nabla, \nabla_{Σ}, \nabla_{\partial Σ}, \partial_{b}$ . Corollary 3.5.
Let $ψ : Γ (\tilde{δ}) \to R$ for some $\tilde{δ} \in (0, δ]$ be sufficiently smooth. Then
$\nabla ψ = \partial_{n} ψ \nabla r + \nabla_{τ} ψ$ on $Γ (\tilde{δ})$ and there are $c, C > 0$ independent of ψ and $\tilde{δ}$ such that $\begin{array}{l} c (| \partial_{n} ψ | + | \nabla_{τ} ψ |) ⩽ | \nabla ψ | ⩽ C (| \partial_{n} ψ | + | \nabla_{τ} ψ |) on Γ (\tilde{δ}), \\ c | \partial_{n} ψ | ⩽ | \nabla r \partial_{r} (ψ \circ \overline{X}) \circ {\overline{X}}^{- 1} | ⩽ C | \partial_{n} ψ | on Γ (\tilde{δ}), \\ c | \nabla_{τ} ψ | ⩽ | \nabla_{Σ} (ψ \circ \overline{X}) \circ {\overline{X}}^{- 1} | ⩽ C | \nabla_{τ} ψ | on Γ (\tilde{δ}) . \end{array}$

It holds $| \nabla ψ |^{2} = | \partial_{n} ψ |^{2} + | \nabla_{τ} ψ |^{2}$ on $Γ (\tilde{δ}, μ_{0})$ .

Set $\overline{Y} : [- δ, δ] \times \partial Σ \times [0, 2 μ_{1}] \times [0, T] \to [- δ, δ] \times Σ \times [0, T] : (r, σ, b, t) \mapsto (r, Y (σ, b), t)$ and $\overline{ψ} : = ψ \circ \overline{X} \circ \overline{Y} |_{(- \tilde{δ}, \tilde{δ}) \times \partial Σ \times [0, 2 μ_{1}] \times [0, T]}$ . Then $\begin{matrix} \tilde{c} (| \nabla_{\partial Σ} \overline{ψ} | + | \partial_{b} \overline{ψ} |) ⩽ | \nabla_{Σ} [ψ \circ \overline{X}] \circ \overline{Y} | ⩽ \tilde{C} (| \nabla_{\partial Σ} \overline{ψ} | + | \partial_{b} \overline{ψ} |) \end{matrix}$ on $(- \tilde{δ}, \tilde{δ}) \times \partial Σ \times [0, 2 μ_{1}] \times [0, T]$ for some $\tilde{c}, \tilde{C} > 0$ independent of $ψ, \tilde{δ}$ .
Analogous assertions hold for ψ defined on $Γ_{t} (\tilde{δ})$ , $t \in [0, T]$ and for ψ defined on open subsets of $Γ (\tilde{δ})$ or $Γ_{t} (\tilde{δ})$ , $t \in [0, T]$ with natural adjustments and uniform constants (w.r.t. ψ, t and the sets). Finally, the assertions are also valid for $R^{m}$ -valued functions, in particular $\begin{matrix} \nabla \vec{ψ} : = {(D_{x} \vec{ψ})}^{⊤} = \nabla r \partial_{n} \vec{ψ} + \nabla_{τ} \vec{ψ} . \end{matrix}$
Proof.
We only consider $ψ : Γ (\tilde{δ}) \to R$ . The case of sufficiently smooth $ψ : Γ_{t} (\tilde{δ}) \to R$ for any $t \in [0, T]$ and the case of other open subsets can be shown with analogous arguments.
Ad 1.
The second equivalence estimate is evident since $0 < \tilde{c} ⩽ | \nabla r | ⩽ \tilde{C}$ due to Theorem 3.2. Moreover, it holds $ψ = (ψ \circ \overline{X}) \circ {\overline{X}}^{- 1} |_{(- \tilde{δ}, \tilde{δ}) \times Σ \times [0, T]}$ . The chain rule yields $\begin{matrix} \nabla ψ |_{(x, t)} \cdot . = d_{x} [ψ (\cdot, t)] = d_{(r, s)} [ψ \circ \overline{X} (\cdot, t)] \circ d_{x} [X {(\cdot, t)}^{- 1}] : R^{N} \to R \end{matrix}$ for all $(x, t) = \overline{X} (r, s, t) \in Γ (\tilde{δ})$ . Here $\begin{array}{l} d_{(r, s)} [ψ \circ \overline{X} (\cdot, t)] : R \times T_{s} Σ \to R : (w, \vec{v}) & \mapsto d_{r} [ψ \circ \overline{X} |_{(\cdot, s, t)}] (w) + d_{s} [ψ \circ \overline{X} |_{(r, \cdot, t)}] (\vec{v}) \\ = \partial_{\tilde{r}} [ψ \circ \overline{X} |_{(\cdot, s, t)}] |_{r} w + \nabla_{Σ} [ψ \circ \overline{X} |_{(r, \cdot, t)}] |_{s} \cdot \vec{v} . \end{array}$ Furthermore, $d_{x} [X {(\cdot, t)}^{- 1}] : R^{N} \to R \times T_{s} Σ : \vec{u} \mapsto (\nabla r \cdot \vec{u}, D_{x} s \vec{u})$ is invertible for all $(x, t) = \overline{X} (r, s, t) \in \overline{Γ (δ)}$ and the operator norm and the one of the inverse is uniformly bounded due to compactness. This yields $\nabla ψ = \partial_{n} ψ \nabla r + \nabla_{τ} ψ$ on $Γ (\tilde{δ})$ and $\begin{matrix} c (| \partial_{n} ψ | + | \nabla_{Σ} (ψ \circ \overline{X}) \circ {\overline{X}}^{- 1} |) ⩽ | \nabla ψ | ⩽ C (| \partial_{n} ψ | + | \nabla_{Σ} (ψ \circ \overline{X}) \circ {\overline{X}}^{- 1} |) on Γ (\tilde{δ}) \end{matrix}$ with $c, C > 0$ independent of $ψ, \tilde{δ}$ . In order to show 1. it remains to prove the last equivalence estimate in the claim. The latter is valid since $D_{x} s$ is uniformly bounded and $D_{x} s {(D_{x} s)}^{⊤}$ is uniformly positive definite on $T_{s (x, t)} Σ$ for all $(x, t) \in \overline{Γ (δ)}$ due to Theorem 3.2. □
Ad 2.
Consequence of 1. and $| \nabla r | = 1$ , $D_{x} s \nabla r = 0$ on $Γ (\tilde{δ}, μ_{0})$ due to Theorem 3.2, 3. □
Ad 3.
The chain rule yields $d_{(σ, b)} [\overline{ψ} (r, \cdot, t)] = d_{Y (σ, b)} [ψ \circ \overline{X} (r, \cdot, t)] \circ d_{(σ, b)} Y : T_{σ} \partial Σ \times R \to R$ for all $(r, σ, b, t) \in (- \tilde{δ}, \tilde{δ}) \times \partial Σ \times [0, 2 μ_{1}] \times [0, T]$ . Here $d_{(σ, b)} Y : T_{σ} \partial Σ \times R \to T_{Y (σ, b)} Σ$ is invertible for all $(σ, b) \in \partial Σ \times [0, 2 μ_{1}]$ and the operator norm and the one of the inverse is uniformly bounded by compactness. Moreover, it holds $\begin{matrix} d_{(σ, b)} [\overline{ψ} (r, \cdot, t)] (\vec{v}, w) = \nabla_{\partial Σ} [\overline{ψ} (r, \cdot, b, t)] |_{σ} \cdot \vec{v} + \partial_{b} [\overline{ψ} (r, σ, \cdot, t)] |_{b} w \end{matrix}$ for all $(\vec{v}, w) \in T_{σ} \partial Σ \times R$ and $d_{Y (σ, b)} [ψ \circ \overline{X} (r, \cdot, t)] (\vec{u}) = \nabla_{Σ} [ψ \circ \overline{X} (r, \cdot, t)] |_{Y (σ, b)} \cdot \vec{u}$ for all $\vec{u} \in T_{Y (σ, b)} Σ$ . This proves the claim. □

4. Model problems

Unless otherwise stated we use real-valued function spaces in this section.

4.1. Some scalar-valued ODE problems on $R$

In this section we recall existence and regularity results needed for ODEs appearing in the inner asymptotic expansion for (AC), cf. Section 5.1.1 below. For the potential $f : R \to R$ in this section we assume (1.1).

4.1.1. The ODE for the optimal profile

The ODE system for the lowest order (cf. Section 5.1.1 below) is $\begin{matrix} (4.1) & - w^{″} + f^{'} (w) = 0, w (0) = 0, lim_{z \to \pm \infty} w (z) = \pm 1 . \end{matrix}$

Theorem 4.1 ([75, Lemma 2.6.1]).

Let f be as in ( 1.1 ). Then ( 4.1 ) has a unique solution $θ_{0} \in C^{2} (R)$ . Moreover, $θ_{0}$ is smooth, $θ_{0}^{'} = \sqrt{2 (f (θ_{0}) - f (- 1))} > 0$ and $\begin{matrix} D_{z}^{k} (θ_{0} \mp 1) (z) = O (e^{- β | z |}) for z \to \pm \infty and all k \in N_{0}, β \in (0, \sqrt{min {f^{″} (\pm 1)}}) . \end{matrix}$

Proof.
The idea is to solve the equivalent first order ODE $\begin{matrix} w^{'} = \sqrt{\int_{- 1}^{w} 2 f^{'} (s) d s}, w (0) = 0 . \end{matrix}$ Note that only ODE-methods and elementary arguments are used. □

We call $θ_{0}$ the optimal profile. A rescaled version will be the typical profile of the solutions for the scalar-valued Allen–Cahn equation with Neumann boundary condition (AC1)–(AC3) across the interface. If f is even, then $θ_{0}$ is even, $θ_{0}^{'}$ is odd and $θ_{0}^{″}$ even etc. For the typical double-well potential $f (u) = \frac{1}{2} {(1 - u^{2})}^{2}$ shown in Fig. 1 one can directly compute that the optimal profile is $θ_{0} = tanh$ , cf. Fig. 4.

Fig. 4.
Typical optimal profile $θ_{0} = tanh$ and the derivative $θ_{0}^{'}$ .
4.1.2. The linearized ODE

The following theorem is concerned with the solvability of the equation which is obtained by linearization of (4.1) at $θ_{0}^{'}$ , i.e. $\begin{matrix} (4.2) & L_{0} w = A in R, w (0) = 0, \end{matrix}$ where we have set $L_{0} : = - \frac{d^{2}}{d z^{2}} + f^{″} (θ_{0})$ . The latter will appear in higher orders in the inner asymptotic expansion for (AC), cf. Section 5.1.1 below.

Theorem 4.2 ([75, Lemma 2.6.2]).

Let $A \in C_{b}^{0} (R)$ and $A^{\pm}$ be constants. Then ( 4.2 ) has a solution $w \in C^{2} (R) \cap C_{b}^{0} (R)$ if and only if $\int_{R} A θ_{0}^{'} d z = 0$ . In that case w is unique. Moreover, if $A (z) - A^{\pm} = O (e^{- β | z |})$ as $z \to \pm \infty$ for some $β \in (0, \sqrt{min {f^{″} (\pm 1)}})$ , then $\begin{matrix} D_{z}^{l} [w - \frac{A^{\pm}}{f^{″} (\pm 1)}] = O (e^{- β | z |}) as z \to \pm \infty, l = 0, 1, 2 . \end{matrix}$

Let $U \subseteq R^{d}$ (any set U is allowed, e.g. a point; role of additional parameters) and $A : R \times U \to R$ , $A^{\pm} : U \to R$ be smooth (i.e. locally smooth extendible) and the following hold uniformly in U: $\begin{matrix} D_{x}^{k} D_{z}^{l} [A (z, \cdot) - A^{\pm}] = O (e^{- β | z |}) as z \to \pm \infty, k = 0, \dots, K, l = 0, \dots, L, \end{matrix}$ for some $β \in (0, \sqrt{min {f^{″} (\pm 1)}})$ and $K, L \in N_{0}$ . Then $w : R \times U \to R$ , where $w (\cdot, x)$ is the solution of ( 4.2 ) for $A (\cdot, x)$ for all $x \in U$ , is also smooth and uniformly in U it holds $\begin{matrix} D_{x}^{k} D_{z}^{l} [w (z, \cdot) - \frac{A^{\pm}}{f^{″} (\pm 1)}] = O (e^{- β | z |}) as z \to \pm \infty, k = 0, \dots, K, l = 0, \dots, L + 2 . \end{matrix}$

For our purpose $A^{\pm} = 0$ will be enough.

Proof.
The result follows from the proof of [75, Lemma 2.6.2]. The idea is to reduce to a first order ODE for the derivative of $w / θ_{0}^{'}$ . In order to show boundedness of the $w \in C^{2} (R)$ in [75] for $A \in C_{b}^{0} (R)$ provided $\int_{R} A θ_{0}^{'} d z = 0$ , one can use $θ_{0}^{'} > 0$ , estimate A roughly in the formula for w in [75] and apply the convergence proof in [75] for the case of constant A there. Note that only ODE-methods and elementary arguments are used. □

4.2. An elliptic problem on $R_{+}^{2}$ with Neumann boundary condition

Let $f : R \to R$ be as in (1.1) and $θ_{0}$ be as in Theorem 4.1. For the contact point expansion for (AC) in any dimension $N ⩾ 2$ (cf. Section 5.1.2 below) we have to solve the following model problem on $R_{+}^{2}$ : For data $G : \overline{R_{+}^{2}} \to R$ , $g : R \to R$ with suitable regularity and exponential decay find a solution $u : \overline{R_{+}^{2}} \to R$ with similar decay to $\begin{array}{l} (4.3) & [- Δ + f^{″} (θ_{0} (R))] u (R, H) = G (R, H) for (R, H) \in R_{+}^{2}, \\ (4.4) & - \partial_{H} u |_{H = 0} (R) = g (R) for R \in R . \end{array}$

In Section 4.2.1 we recall existence and uniqueness assertions for weak solutions from [5]. For the proofs see [5, Section 2.4.1]. Moreover, in [5, Section 2.4.2] exponential decay estimates were proven via ordinary differential inequality arguments. Here we proceed differently. In Section 4.2.2 we introduce a functional analytic setting with several types of exponentially weighted Sobolev spaces in order to have solution operators for (4.3)–(4.4). The rough idea is always to multiply the equation with the weights, use the product rule and known isomorphisms.

The framework with exponentially weighted Sobolev spaces helps with additional independent variables and simplifies the induction procedure needed in the asymptotic expansion in Section 5.1 below. Note that in the 2D-case an induction is not necessary and hence was not carried out in [5]. Moreover, the isomorphism property can be used to solve via the Implicit Function Theorem a corresponding model problem appearing in the sharp interface limit for an Allen–Cahn equation with a nonlinear Robin boundary condition designed to approximate (MCF) with a constant α-contact angle for α close to $\frac{π}{2}$ . See [66] or [6] for details.

4.2.1. Weak solutions and regularity

The definition of a weak solution is canonical:

Definition 4.3 ([5, Definition 2.7]).

Let $G \in L^{2} (R_{+}^{2})$ and $g \in L^{2} (R)$ . Then $u \in H^{1} (R_{+}^{2})$ is called weak solution of (4.3)–(4.4) if for all $φ \in H^{1} (R_{+}^{2})$ it holds that $\begin{matrix} a (u, φ) : = \int_{R_{+}^{2}} \nabla u \cdot \nabla φ + f^{″} (θ_{0} (R)) u φ d (R, H) = \int_{R_{+}^{2}} G φ d (R, H) + \int_{R} g (R) φ |_{H = 0} (R) d R . \end{matrix}$

Regarding weak solutions we have the following theorem:

Theorem 4.4 ([5, Theorem 2.9]).

Let $G \in L^{2} (R_{+}^{2})$ and $g \in L^{2} (R)$ . Then it holds:

$a : H^{1} (R_{+}^{2}) \times H^{1} (R_{+}^{2}) \to R$ is not coercive.

If $G (\cdot, H), g ⊥ θ_{0}^{'}$ for a.e. $H > 0$ in $L^{2} (R)$ , then there is a weak solution u such that $u (\cdot, H) ⊥ θ_{0}^{'}$ for a.e. $H > 0$ and it holds $‖ u ‖_{H^{1} (R_{+}^{2})} ⩽ C (‖ G ‖_{L^{2} (R_{+}^{2})} + ‖ g ‖_{L^{2} (R)})$ .

Weak solutions are unique.

If $G θ_{0}^{'} \in L^{1} (R_{+}^{2})$ and u is a weak solution with $\partial_{H} u θ_{0}^{'} \in L^{1} (R_{+}^{2})$ , then the following compatibility condition holds: $\begin{matrix} (4.5) & \int_{R_{+}^{2}} G (R, H) θ_{0}^{'} (R) d (R, H) + \int_{R} g (R) θ_{0}^{'} (R) d R = 0 . \end{matrix}$

If $G θ_{0}^{'} \in L^{1} (R_{+}^{2})$ , then $\tilde{G} (H) : = {(G (\cdot, H), θ_{0}^{'})}_{L^{2} (R)}$ is well-defined for a.e. $H > 0$ and $\tilde{G} \in L^{1} (R_{+}) \cap L^{2} (R_{+})$ . Moreover, we have the decomposition $\begin{matrix} (4.6) & G = \tilde{G} (H) \frac{θ_{0}^{'} (R)}{‖ θ_{0}^{'} ‖_{L^{2} (R)}^{2}} + G^{⊥} (R, H), g = {(g, θ_{0}^{'})}_{L^{2} (R)} \frac{θ_{0}^{'} (R)}{‖ θ_{0}^{'} ‖_{L^{2} (R)}^{2}} + g^{⊥} (R) \end{matrix}$ for some $G^{⊥} \in L^{2} (R_{+}^{2}), g^{⊥} \in L^{2} (R)$ with $G^{⊥} (\cdot, H), g ⊥ θ_{0}^{'}$ in $L^{2} (R)$ for a.e. $H > 0$ .

If $‖ G (\cdot, H) ‖_{L^{2} (R)} ⩽ C e^{- ν H}$ for a.e. $H > 0$ and a constant $ν > 0$ , then $G θ_{0}^{'} \in L^{1} (R_{+}^{2})$ . Let $\tilde{G}$ be defined as in 5. and the compatibility condition ( 4.5 ) hold. Then $\begin{matrix} (4.7) & u_{1} (R, H) : = - \int_{H}^{\infty} \int_{\tilde{H}}^{\infty} \tilde{G} (\hat{H}) d \hat{H} d \tilde{H} \frac{θ_{0}^{'} (R)}{‖ θ_{0}^{'} ‖_{L^{2} (R)}^{2}} \end{matrix}$ is well-defined for a.e. $(R, H) \in R_{+}^{2}$ , it holds $u_{1} \in W_{1}^{2} (R_{+}^{2}) \cap H^{2} (R_{+}^{2})$ and $u_{1}$ is a weak solution of ( 4.3 )–( 4.4 ) for $G - G^{⊥}, g - g^{⊥}$ from ( 4.6 ) instead of G, g.

Here the spectral properties of the linear operator $L_{0} : H^{2} (R) \subseteq L^{2} (R) \to L^{2} (R) : u \mapsto [- \frac{d^{2}}{d R^{2}} + f^{″} (θ_{0})] u$ are important. See [66, Lemma 4.2] or [5, Lemma 2.5] for the latter.

In Theorem 4.4, 6. weaker conditions on G are enough, cf. Section 4.2.2. The point is included for aesthetic reasons. Altogether the following existence theorem for weak solutions is obtained:

Corollary 4.5 ([5, Corollary 2.10]).

Let $g \in L^{2} (R)$ and $G \in L^{2} (R_{+}^{2})$ be such that $‖ G (\cdot, H) ‖_{L^{2} (R)} ⩽ C e^{- ν H}$ f.a.e. $H > 0$ and some $ν > 0$ . Let ( 4.5 ) hold. Then there is a unique weak solution of ( 4.3 )–( 4.4 ).

Let $k \in N_{0}$ and $u \in H^{1} (R_{+}^{2})$ be a weak solution of ( 4.3 )–( 4.4 ) for $G \in H^{k} (R_{+}^{2})$ and $g \in H^{k + \frac{1}{2}} (R)$ . Then $u \in H^{k + 2} (R_{+}^{2}) ↪ C^{k, γ} (\overline{R_{+}^{2}})$ for all $γ \in (0, 1)$ and it holds $\begin{matrix} ‖ u ‖_{H^{k + 2} (R_{+}^{2})} ⩽ C_{k} (‖ G ‖_{H^{k} (R_{+}^{2})} + ‖ g ‖_{H^{k + \frac{1}{2}} (R)} + ‖ u ‖_{H^{1} (R_{+}^{2})}) . \end{matrix}$

4.2.2. Solution operators in exponentially weighted spaces

In the following the superscript “⊥” always means $u (\cdot, H) ⊥ θ_{0}^{'}$ in $L^{2} (R)$ for a.e. $H \in R_{+}$ , if $u \in L^{2} (R_{+}^{2})$ , and $u ⊥ θ_{0}^{'}$ if $u \in L^{2} (R)$ . The symbol “∥” has the same meaning with “⊥” replaced by “∥”. Moreover, recall the exponentially weighted spaces from Definition 2.4.

Theorem 4.6 (Solution Operators for Decay inH).

For $γ ⩾ 0$ small the operator $\begin{matrix} L_{\frac{π}{2}} : = (- Δ + f^{″} (θ_{0} (R)), - \partial_{H} |_{H = 0}) : H_{(0, γ)}^{2, ⊥} (R_{+}^{2}) \to L_{(0, γ)}^{2, ⊥} (R_{+}^{2}) \times H^{\frac{1}{2}, ⊥} (R) \end{matrix}$ is well-defined and invertible. Moreover, for small $γ ⩾ 0$ the operator norm of $L_{\frac{π}{2}}^{- 1}$ is uniformly bounded in the corresponding spaces.

For all $γ \in (0, \overline{γ}]$ and any $\overline{γ} > 0$ the operator $\begin{matrix} L_{\frac{π}{2}} : H_{(0, γ)}^{2, ∥} (R_{+}^{2}) \to {(G, g) \in L_{(0, γ)}^{2, ∥} (R_{+}^{2}) \times H^{\frac{1}{2}, ∥} (R) : \int_{R_{+}^{2}} G θ_{0}^{'} + \int_{R} g θ_{0}^{'} = 0} \end{matrix}$ is well-defined, invertible and the norm of $L_{\frac{π}{2}}^{- 1}$ is bounded by $C_{\overline{γ}} (1 + \frac{1}{γ^{2}})$ for all $γ \in (0, \overline{γ}]$ .

For $γ > 0$ small $\begin{matrix} L_{\frac{π}{2}} : H_{(0, γ)}^{2} (R_{+}^{2}) \to {(G, g) \in L_{(0, γ)}^{2} (R_{+}^{2}) \times H^{\frac{1}{2}} (R) : \int_{R_{+}^{2}} G θ_{0}^{'} + \int_{R} g θ_{0}^{'} = 0} \end{matrix}$ is well-defined, invertible and the norm of $L_{\frac{π}{2}}^{- 1}$ is bounded by $C (1 + \frac{1}{γ^{2}})$ for small $γ > 0$ .

Proof. Ad 1.
$L_{\frac{π}{2}}$ is well-defined in the spaces because of Lemma 2.5, 2., 5. and since the orthogonality property can be shown via integration by parts as well as by differentiating the orthogonality condition for u with respect to H. In the case $γ = 0$ invertibility follows from Theorem 4.4, 2.–3. and Corollary 4.5, 2. Now let $γ > 0$ . In order to solve $L_{\frac{π}{2}} u = (G, g)$ we make the ansatz $u = e^{- γ H} v$ with $v \in H^{2, ⊥} (R_{+}^{2})$ . By computing derivatives of u we obtain equations we want to solve for v. Note that the exponential factor does not destroy the orthogonality property. It holds $\begin{matrix} \partial_{H} u = - γ e^{- γ H} v + e^{- γ H} \partial_{H} v and \partial_{H}^{2} u = γ^{2} e^{- γ H} v - 2 γ e^{- γ H} \partial_{H} v + e^{- γ H} \partial_{H}^{2} v . \end{matrix}$ Therefore we consider $\begin{matrix} (4.8) & L_{\frac{π}{2}} v + N_{γ} v = (G e^{γ H}, g), where N_{γ} v : = (- γ^{2} v + 2 γ \partial_{H} v, γ v |_{H = 0}) . \end{matrix}$ Here $N_{γ}$ is a bounded linear operator from $H^{2, ⊥} (R_{+}^{2})$ to $L^{2, ⊥} (R_{+}^{2}) \times H^{\frac{1}{2}, ⊥} (R)$ and the operator norm is estimated by $C (γ + γ^{2})$ . Hence a Neumann series argument yields that $L_{\frac{π}{2}} + N_{γ}$ is invertible in those spaces for small γ and the norm of the inverse is bounded uniformly. Let $(G, g)$ be in $L_{(0, γ)}^{2, ⊥} (R_{+}^{2}) \times H^{\frac{1}{2}} (R)$ . Then we obtain for small $γ > 0$ a unique $v \in H^{2, ⊥} (R_{+}^{2})$ that solves (4.8). The above computations yield that $u : = e^{- γ H} v \in H_{(0, γ)}^{2, ⊥} (R_{+}^{2})$ is a solution of $L_{\frac{π}{2}} u = (G, g)$ and that it is unique. Finally, we have the estimate $\begin{matrix} ‖ u ‖_{H_{(0, γ)}^{2, ⊥} (R_{+}^{2})} = ‖ v ‖_{H^{2, ⊥} (R_{+}^{2})} ⩽ C {‖ (G e^{γ H}, g) ‖}_{L^{2, ⊥} (R_{+}^{2}) \times H^{\frac{1}{2}} (R)} = C {‖ (G, g) ‖}_{L_{(0, γ)}^{2, ⊥} (R_{+}^{2}) \times H^{\frac{1}{2}} (R)}, \end{matrix}$ where $C > 0$ is independent of $γ > 0$ small. □
Ad 2.
Let $γ > 0$ . Then $u \in H_{(0, γ)}^{2, ∥} (R_{+}^{2})$ if and only if $\begin{matrix} e^{γ H} u \in H^{2} (R_{+}^{2}) and u (R, H) = {(u (\cdot, H), θ_{0}^{'})}_{L^{2} (R)} \frac{θ_{0}^{'} (R)}{‖ θ_{0}^{'} ‖_{L^{2} (R)}^{2}} almost everywhere . \end{matrix}$ Since $H^{2} (R_{+}^{2}) ↪ H^{2} (R_{+}, L^{2} (R))$ and because multiplication with $θ_{0}^{'}$ is a bounded linear operator from $L^{2} (R)$ to $R$ , it follows that $u \in H_{(0, γ)}^{2, ∥} (R_{+}^{2})$ is equivalent to $u (R, H) = \tilde{u} (H) θ_{0}^{'} (R)$ for a.a. $(R, H) \in R_{+}^{2}$ for some $\tilde{u} \in H_{(γ)}^{2} (R_{+})$ . The operator $L_{\frac{π}{2}}$ acts as $\begin{matrix} L_{\frac{π}{2}} u = (- \partial_{H}^{2} \tilde{u} (H) θ_{0}^{'} (R), - \partial_{H} \tilde{u} (0) θ_{0}^{'} (R)) \in L_{(0, γ)}^{2, ∥} (R_{+}^{2}) \times H^{\frac{1}{2}, ∥} (R) . \end{matrix}$ Additionally, the compatibility condition (4.5) holds because of Theorem 4.4, 4. The latter could also be directly computed here. Altogether, $L_{\frac{π}{2}}$ is well-defined in the spaces. Moreover, let $(G, g)$ be in the space $L_{(0, γ)}^{2, ∥} (R_{+}^{2}) \times H^{\frac{1}{2}, ∥} (R)$ and let the compatibility condition (4.5) hold. Then $G = \tilde{G} (H) θ_{0}^{'} (R)$ , $g = \tilde{g} θ_{0}^{'}$ for some $\tilde{G} \in L_{(γ)}^{2} (R_{+})$ and $\tilde{g} \in R$ . By Lemma 2.5, 6.–7. it holds $\begin{matrix} \tilde{u} : = - \int_{\cdot}^{\infty} \int_{\hat{H}}^{\infty} \tilde{G} (\overline{H}) d \overline{H} d \hat{H} \in H_{(γ)}^{2} (R_{+}) \end{matrix}$ with $\partial_{H} \tilde{u} = \int_{\cdot}^{\infty} \tilde{G} (\hat{H}) d \hat{H}$ and $\partial_{H}^{2} \tilde{u} = - \tilde{G}$ as well as $‖ \tilde{u} ‖_{H_{(γ)}^{2} (R_{+})} ⩽ c_{\overline{γ}} (1 + \frac{1}{γ^{2}}) ‖ \tilde{G} ‖_{L_{(γ)}^{2} (R_{+})}$ for all $γ \in (0, \overline{γ}]$ , where $\overline{γ} > 0$ is fixed. Therefore $u_{1} : = \tilde{u} (H) θ_{0}^{'} (R) \in H_{(0, γ)}^{2} (R_{+}^{2})$ solves $\begin{array}{l} [- Δ + f^{″} (θ_{0})] u_{1} = - \partial_{H}^{2} \tilde{u} θ_{0}^{'} = G, \\ - \partial_{H} u_{1} |_{H = 0} = - \int_{0}^{\infty} \tilde{G} (\hat{H}) d \hat{H} θ_{0}^{'} = \tilde{g} θ_{0}^{'} = g, \end{array}$ where the last equality follows from the compatibility condition (4.5). Hence $u_{1}$ is a solution of $L_{\frac{π}{2}} u_{1} = (G, g)$ and it is unique because of Theorem 4.4, 3. Moreover, we have $\begin{matrix} ‖ u_{1} ‖_{H_{(0, γ)}^{2} (R_{+}^{2})} ⩽ C ‖ \tilde{u} ‖_{H_{(γ)}^{2} (R_{+})} ⩽ C c_{\overline{γ}} (1 + \frac{1}{γ^{2}}) ‖ \tilde{G} ‖_{L_{(γ)}^{2} (R_{+})} ⩽ C_{\overline{γ}} (1 + \frac{1}{γ^{2}}) ‖ G ‖_{L_{(0, γ)}^{2} (R_{+}^{2})} . \end{matrix}$ Altogether this proves the claim. □
Ad 3.
Via (4.6) we have isomorphic splitting operators from $H_{(0, γ)}^{k} (R_{+}^{2})$ onto the direct sum $H_{(0, γ)}^{k, ⊥} (R_{+}^{2}) \oplus H_{(0, γ)}^{k, ∥} (R_{+}^{2})$ for all $k \in N_{0}$ and the operator norms for fixed k are estimated by a constant independent of $γ \in (0, \overline{γ}]$ . Therefore the claim follows from 1. and 2. □
Theorem 4.7 (Solution Operators for Decay in $(R, H)$ ).

Let $\overline{γ} > 0$ be such that Theorem 4.6 , 3. holds for $γ \in (0, \overline{γ}]$ . Then there is a non-decreasing $\overline{β} : (0, \overline{γ}] \to (0, \infty)$ such that $\begin{matrix} L_{\frac{π}{2}} : H_{(β, γ)}^{2} (R_{+}^{2}) \to Y_{(β, γ)} : = {(G, g) \in L_{(β, γ)}^{2} (R_{+}^{2}) \times H_{(β)}^{\frac{1}{2}} (R) : \int_{R_{+}^{2}} G θ_{0}^{'} + \int_{R} g θ_{0}^{'} = 0} \end{matrix}$ is an isomorphism for all $β \in [0, \overline{β} (γ)]$ and the operator norm of the inverse is bounded by $\tilde{C} (1 + \frac{1}{γ^{2}})$ with $\tilde{C}$ independent of $(β, γ)$ .

Proof.
The idea is similar as in the proof of Theorem 4.6, 1. $L_{\frac{π}{2}}$ is well-defined in the spaces due to Theorem 4.4, 4. In order to solve $L_{\frac{π}{2}} u = (G, g) \in Y_{(β, γ)}$ , we make the ansatz $u = e^{- β \hat{η} (R)} v$ for $v \in H_{(0, γ)}^{2} (R_{+}^{2})$ , where $\hat{η} : R \to R$ is as in Definition 2.4, 3. We compute $\begin{array}{l} \partial_{R} u = e^{- β \hat{η} (R)} [\partial_{R} v - β {\hat{η}}^{'} (R) v], \\ \partial_{R}^{2} u = e^{- β \hat{η} (R)} [\partial_{R}^{2} v - 2 β {\hat{η}}^{'} (R) \partial_{R} v + v (β^{2} {\hat{η}}^{'} {(R)}^{2} - β {\hat{η}}^{″} (R))] . \end{array}$ Therefore for $v \in H_{(0, γ)}^{2} (R_{+}^{2})$ we consider the equation $\begin{matrix} (4.9) & L_{\frac{π}{2}} v + (N_{(β, γ)} v, 0) = e^{β \hat{η} (R)} (G, g), N_{(β, γ)} v : = 2 β {\hat{η}}^{'} \partial_{R} v - v (β^{2} {({\hat{η}}^{'})}^{2} - β {\hat{η}}^{″}) . \end{matrix}$ There is a problem with the compatibility condition (4.5) here. For $L_{\frac{π}{2}} v$ the latter is valid by Theorem 4.4, 4., but for $(N_{(β, γ)} v, 0)$ and $e^{β \hat{η} (R)} (G, g)$ it does not hold necessarily. Therefore we enforce the condition on both sides artificially and look at the adjusted equation $\begin{array}{l} (4.10) & L_{\frac{π}{2}} v + {\tilde{N}}_{(β, γ)} v = (\overline{G}, \overline{g}), \\ {\tilde{N}}_{(β, γ)} v : = (N_{(β, γ)} v, - [\int_{R_{+}^{2}} N_{(β, γ)} v θ_{0}^{'}] \frac{θ_{0}^{'}}{‖ θ_{0}^{'} ‖_{L^{2} (R)}^{2}}), \\ (\overline{G}, \overline{g}) : = e^{β \hat{η} (R)} (G, g) - (0, [\int_{R_{+}^{2}} e^{β \hat{η} (R)} G θ_{0}^{'} + \int_{R} e^{β \hat{η} (R)} g θ_{0}^{'}] \frac{θ_{0}^{'}}{‖ θ_{0}^{'} ‖_{L^{2} (R)}^{2}}) . \end{array}$

In order to solve (4.10), we observe that ${\tilde{N}}_{(β, γ)}$ is a bounded linear operator from $H_{(0, γ)}^{2} (R_{+}^{2})$ to $Y_{(0, γ)}$ with norm estimated by $C (β + β^{2})$ for all $β ⩾ 0$ and $γ \in (0, \overline{γ}]$ . Moreover, Theorem 4.6, 3. yields that $L_{\frac{π}{2}}$ is an isomorphism in these spaces and that the inverse is bounded by $\overline{C} (1 + \frac{1}{γ^{2}})$ for all $γ \in (0, \overline{γ}]$ . We choose $\overline{β} = \overline{β} (γ)$ such that $C (\overline{β} + {\overline{β}}^{2}) ⩽ 1 / [2 \overline{C} (1 + \frac{1}{γ^{2}})]$ and such that $\overline{β} : (0, \overline{γ}] \to (0, \infty)$ is non-decreasing. Then a Neumann series argument yields that $L_{\frac{π}{2}} + {\tilde{N}}_{(β, γ)}$ is invertible from $H_{(0, γ)}^{2} (R_{+}^{2})$ onto $Y_{(0, γ)}$ for all $β \in [0, \overline{β} (γ)]$ , $γ \in (0, \overline{γ}]$ and the norm of the inverse is bounded by $2 \overline{C} (1 + \frac{1}{γ^{2}})$ .

Now let $β \in [0, \overline{β} (γ)]$ , $γ \in (0, \overline{γ}]$ and $v \in H_{(0, γ)}^{2} (R_{+}^{2})$ solve (4.10) for $(G, g) \in Y_{(β, γ)}$ . Then $u : = e^{- β \hat{η} (R)} v \in H_{(β, γ)}^{2} (R_{+}^{2})$ is a solution of $\begin{matrix} L_{\frac{π}{2}} u = (G, g) + (0, \frac{e^{- β \hat{η} (R)} θ_{0}^{'}}{‖ θ_{0}^{'} ‖_{L^{2} (R)}^{2}} [\int_{R_{+}^{2}} (N_{(β, γ)} v + e^{β \hat{η} (R)} G) θ_{0}^{'} + \int_{R} e^{β \hat{η} (R)} g θ_{0}^{'}]) . \end{matrix}$ The compatibility condition (4.5) holds for $L_{\frac{π}{2}} u$ and $(G, g)$ . Since $\int_{R} e^{- β \hat{η} (R)} θ_{0}^{'} {(R)}^{2} d R$ is positive, it follows that the second term is zero for the solution, i.e. u is a solution of $L_{\frac{π}{2}} u = (G, g)$ . By construction or alternatively by Theorem 4.4, 3. the solution is unique and we have the estimate $\begin{matrix} ‖ u ‖_{H_{(β, γ)}^{2} (R_{+}^{2})} = ‖ v ‖_{H_{(0, γ)}^{2} (R_{+}^{2})} ⩽ 2 \overline{C} (1 + \frac{1}{γ^{2}}) {‖ (\overline{G}, \overline{g}) ‖}_{Y_{(0, γ)}} ⩽ \tilde{C} (1 + \frac{1}{γ^{2}}) {‖ (G, g) ‖}_{Y_{(β, γ)}} \end{matrix}$ with $\tilde{C} > 0$ independent of $β, γ$ and the functions. This proves the theorem. □
Theorem 4.8 (Solution Operators for Higher Regularity).

Let $\overline{β} : (0, \overline{γ}] \to (0, \infty)$ and $\overline{γ} > 0$ be as in Theorem 4.7 . Then for all $k \in N_{0}$ , $γ \in (0, \overline{γ}]$ and $β \in [0, \overline{β} (γ)]$ it follows that $\begin{matrix} L_{\frac{π}{2}} : H_{(β, γ)}^{k + 2} (R_{+}^{2}) \to Y_{(β, γ)}^{k} : = {(G, g) \in H_{(β, γ)}^{k} (R_{+}^{2}) \times H_{(β)}^{k + \frac{1}{2}} (R) : \int_{R_{+}^{2}} G θ_{0}^{'} + \int_{R} g θ_{0}^{'} = 0} \end{matrix}$ is invertible and the operator norm of the inverse is bounded by $C (k) {(1 + \frac{1}{γ^{2}})}^{k + 1}$ .

Proof.
$L_{\frac{π}{2}}$ is a well-defined, bounded linear operator in the above spaces. Let $(G, g) \in Y_{(β, γ)}^{k}$ . Then by Theorem 4.7 there is a unique $u \in H_{(β, γ)}^{2} (R_{+}^{2})$ that solves $L_{\frac{π}{2}} u = (G, g)$ . By regularity theory, cf. Corollary 4.5, 2., it follows that $u \in H^{k + 2} (R_{+}^{2})$ .

Now we show $\partial_{R}^{l} u \in H_{(β, γ)}^{2} (R_{+}^{2})$ for all $l = 1, \dots, k$ and suitable estimates. To this end we apply $\partial_{R}^{l}$ to the equation and get $\begin{matrix} L_{\frac{π}{2}} \partial_{R}^{l} u = (\partial_{R}^{l} G - \sum_{j = 1}^{l} (\begin{matrix} l \\ j \end{matrix}) \partial_{R}^{j} (f^{″} (θ_{0})) \partial_{R}^{l - j} u, \partial_{R}^{l} g) = : (G_{l}, g_{l}) . \end{matrix}$ First we consider $l = 1$ . It holds $G_{1} = \partial_{R} G - \partial_{R} (f^{″} (θ_{0})) u \in L_{(β, γ)}^{2} (R_{+}^{2})$ and $g_{1} \in H_{(β)}^{\frac{1}{2}} (R)$ . Moreover, due to Theorem 4.4, 4. the compatibility condition (4.5) holds for $(G_{1}, g_{1})$ . Therefore Theorem 4.7 and the uniqueness of solutions in Theorem 4.4, 3. implies $\partial_{R} u \in H_{(β, γ)}^{2} (R_{+}^{2})$ and $\begin{array}{l} ‖ \partial_{R} u ‖_{H_{(β, γ)}^{2} (R_{+}^{2})} & ⩽ \tilde{C} (1 + \frac{1}{γ^{2}}) {‖ (G_{1}, g_{1}) ‖}_{Y_{(β, γ)}} \\ ⩽ C (1 + \frac{1}{γ^{2}}) (‖ G ‖_{H_{(β, γ)}^{1} (R_{+}^{2})} + ‖ g ‖_{H_{(β)}^{\frac{3}{2}} (R)} + ‖ u ‖_{L_{(β, γ)}^{2} (R_{+}^{2})}), \end{array}$ where $‖ u ‖_{L_{(β, γ)}^{2} (R_{+}^{2})} ⩽ \tilde{C} (1 + \frac{1}{γ^{2}}) ‖ (G, g) ‖_{Y_{(β, γ)}}$ and we used the product rule to rewrite and estimate $e^{β \hat{η} (R)} \partial_{R} g$ . This shows the case $l = 1$ . By mathematical induction on l it follows that $\partial_{R}^{l} u \in H_{(β, γ)}^{2} (R_{+}^{2})$ for $l = 1, \dots, k$ and $‖ \partial_{R}^{l} u ‖_{H_{(β, γ)}^{2} (R_{+}^{2})} ⩽ C (k) {(1 + \frac{1}{γ^{2}})}^{k + 1}$ .

The remaining assertions and estimates will be shown by differentiating and rearranging the first equation in $L_{\frac{π}{2}} u = (G, g)$ . For $l = 0, \dots, k$ we have $\begin{matrix} \partial_{R}^{l} \partial_{H}^{2} u = - \partial_{R}^{l} [- \partial_{R}^{2} u + f^{″} (θ_{0} (R)) u - G] . \end{matrix}$ For $k = 1$ and $l = 1$ this implies $\partial_{H}^{2} u \in H_{(β, γ)}^{1} (R_{+}^{2})$ together with a suitable estimate. Hence in the case $k = 1$ we are done. Now let $k ⩾ 2$ . Then we obtain $\partial_{R}^{l} \partial_{H}^{2} u \in H_{(β, γ)}^{2} (R_{+}^{2})$ for all $l = 0, \dots, k - 2$ with appropriate estimates. This also shows the case $k = 2$ . Now let $k ⩾ 3$ . Applying $\partial_{H}^{2}$ to the above equation and similar arguments as before yield $\partial_{H}^{4} u \in H_{(β, γ)}^{1} (R_{+}^{2})$ in the case $k = 3$ and $\partial_{R}^{l} \partial_{H}^{4} u \in H_{(β, γ)}^{2} (R_{+}^{2})$ for all $k ⩾ 4$ and $l = 0, \dots, k - 4$ . Additionally, one also obtains suitable estimates. One can complete the argument with an induction proof. □

4.3. Some vector-valued ODE problems on $R$

The structure of this section is similar to Section 4.1 which is the analogue in the scalar case. We consider vector-valued ODEs appearing in the inner asymptotic expansion of (vAC) at a two-phase transition (cf. Section 5.2.1 below) and also the linear operator belonging to a linearized ODE. In order to solve the linearized ODE, we will assume that the kernel of the linearization is 1-dimensional. The latter is fulfilled for a typical potential, cf. the example in Remark 4.11 below.

Let $W : R^{m} \to R$ be as in Definition 1.1 and ${\vec{u}}_{\pm}$ be any distinct pair in ${\vec{a}, \vec{b}}$ or ${{\vec{x}}_{1}, {\vec{x}}_{3}, {\vec{x}}_{5}}$ , respectively. From now on, we fix ${\vec{u}}_{\pm}$ .

4.3.1. The nonlinear ODE

The nonlinear ODE problem in the lowest order (cf. Section 5.2.1 below) is the following: Find $\vec{u} : R \to R^{m}$ smooth with suitable decay such that $\begin{matrix} (4.11) & - {\vec{u}}^{″} + \nabla W (\vec{u}) = 0 on R, lim_{z \to \pm \infty} \vec{u} (z) = {\vec{u}}_{\pm} . \end{matrix}$

Theorem 4.9 ([54, Section 2.1], [22, Section 2]).

Let m, W, ${\vec{u}}_{\pm}$ be as above and let $λ > 0$ be such that $D^{2} W ({\vec{u}}_{\pm}) ⩾ λ I$ . Then there is a smooth solution $\vec{u} : R \to R^{m}$ to ( 4.11 ) such that $\begin{matrix} \partial_{z}^{k} [\vec{u} - {\vec{u}}_{\pm}] (z) = O (e^{- β | z |}) for z \to \pm \infty and all k \in N_{0}, β \in (0, \sqrt{λ / 2}) . \end{matrix}$ Moreover, $\vec{u}$ can be chosen $R_{{\vec{u}}_{-}, {\vec{u}}_{+}}$ -odd, i.e. $\vec{u} (- \cdot) = R_{{\vec{u}}_{-}, {\vec{u}}_{+}} \vec{u}$ with $R_{{\vec{u}}_{-}, {\vec{u}}_{+}}$ as in Definition 1.1 .

Remark 4.10.

From now on, we fix a $R_{{\vec{u}}_{-}, {\vec{u}}_{+}}$ -odd solution and simply denote it by ${\vec{θ}}_{0}$ .

The proof relies on minimizing an energy over an appropriate set (see below). Similar to Bronsard, Gui, Schatzman [22, Section 2], where transitions between two minima in the triple-well case is considered, it should be possible to determine the qualitative behaviour of the set of 1D-minimizers for both types of W in Definition 1.1 precisely. E.g. in the triple-well case the 1D-minimizers are trapped in the smaller sector between ${\vec{u}}_{-}$ and ${\vec{u}}_{+}$ . But this is not needed here.

In Bronsard, Gui, Schatzman [22, Section 4] the existence of a solution of the two-dimensional problem $\begin{matrix} - Δ \vec{u} + \nabla W (\vec{u}) = 0 in R^{2} \end{matrix}$ for $\vec{u} : R^{2} \to R^{2}$ with suitable asymptotic behaviour and symmetries is shown for a class of triple-well potentials W (including the class used in the present paper). Such a function is the natural candidate for the lowest order in an expansion of the vector-valued Allen–Cahn equation at a three-phase transition (i.e. a triple junction for multiphase mean curvature flow). Note that in Alikakos, Fusco [10] such a solution was obtained with different methods and in more generality.

Proof of Theorem 4.9.
Let $ξ : R \to R$ be smooth and odd such that $ξ (z) = sign (z)$ for $| z | ⩾ 1$ . Moreover, we define $\begin{matrix} (4.12) & Ξ ({\vec{u}}_{-}, {\vec{u}}_{+}) : = \frac{1}{2} ({\vec{u}}_{-} + {\vec{u}}_{+}) + ξ \frac{1}{2} ({\vec{u}}_{+} - {\vec{u}}_{-}) . \end{matrix}$ Then [54, Section 2.1] for potentials W as in Definition 1.1, 1. and [22, Section 2] for triple-well potentials W in Definition 1.1, 2., respectively, yield that $\begin{matrix} E : Ξ ({\vec{u}}_{-}, {\vec{u}}_{+}) + H^{1} {(R)}^{m} \to R : \vec{u} \mapsto \int_{R} \frac{1}{2} {| {\vec{u}}^{'} |}^{2} + W (\vec{u}) d z \end{matrix}$ admits a global minimizer $\vec{u}$ that satisfies $\vec{u} \in C^{3} {(R)}^{m} \cap (H^{2} {(R)}^{m} + Ξ ({\vec{u}}_{-}, {\vec{u}}_{+}))$ and is $R_{{\vec{u}}_{-}, {\vec{u}}_{+}}$ -odd. Moreover, $\vec{u}$ satisfies (4.11) and $\begin{matrix} \partial_{z}^{k} [\vec{u} - {\vec{u}}_{\pm}] (z) = O (e^{- β | z |}) for z \to \pm \infty and k = 0, 1, 2, 3, β \in (0, \sqrt{λ / 2}) . \end{matrix}$ Furthermore, one obtains $\vec{u} \in C^{k + 1} {(R)}^{m} \cap (H^{k} {(R)}^{m} + Ξ ({\vec{u}}_{-}, {\vec{u}}_{+}))$ for all $k \in N$ , $k ⩾ 2$ and the decay properties by induction and differentiating the equation. □

4.3.2. The linearized operator

We look at the operator obtained by linearization of the left hand side of the ODE (4.11) at ${\vec{θ}}_{0}$ from Remark 4.10, i.e. $\begin{matrix} (4.13) & {\overset{ˇ}{L}}_{0} : H^{2} {(R, K)}^{m} \subseteq L^{2} {(R, K)}^{m} \to L^{2} {(R, K)}^{m} : \vec{u} \mapsto {\overset{ˇ}{L}}_{0} \vec{u} : = [- \frac{d^{2}}{d z^{2}} + D^{2} W ({\vec{θ}}_{0})] \vec{u} . \end{matrix}$

Remark 4.11 (Assumption $dim ker {\overset{ˇ}{L}}_{0} = 1$ ).

${\vec{θ}}_{0}^{'}$ is an element of $ker {\overset{ˇ}{L}}_{0}$ and obviously ${\vec{θ}}_{0}^{'} ≢ 0$ . In order to have a spectral gap property that is needed for solving the vector-valued linearized ODE and the vector-valued $R_{+}^{2}$ -model problem in the next sections, we assume $dim ker {\overset{ˇ}{L}}_{0} = 1$ (this is independent of $K$ since $D^{2} W ({\vec{θ}}_{0})$ is real-valued). This is reasonable, cf. [54, Section 3.4] for a typical triple-well potential that fulfils this. Note that the assumption should be stable under suitable “small” perturbations of the potential W due to the upper continuity of the nullity index for (semi-)Fredholm operators, cf. Kato [51, Theorem 5.22].

Lemma 4.12.
Let ${\overset{ˇ}{L}}_{0}$ be as above and $K = R$ or $C$ . We denote with $σ_{e} ({\overset{ˇ}{L}}_{0})$ and $σ_{d} ({\overset{ˇ}{L}}_{0})$ the essential and discrete spectrum of ${\overset{ˇ}{L}}_{0}$ , respectively. Then ${\overset{ˇ}{L}}_{0}$ is self-adjoint, ${\overset{ˇ}{L}}_{0} ⩾ 0$ and $\begin{matrix} σ_{e} ({\overset{ˇ}{L}}_{0}) = [min {σ (D^{2} W ({\vec{u}}_{\pm}))}, \infty) . \end{matrix}$ In particular $σ_{d} ({\overset{ˇ}{L}}_{0}) \subset [0, min {σ (D^{2} W ({\vec{u}}_{\pm}))})$ . Moreover, if $dim ker {\overset{ˇ}{L}}_{0} = 1$ , then with ${(ker {\overset{ˇ}{L}}_{0})}^{⊥} : = {\vec{w} \in L^{2} {(R, C)}^{m} : {(\vec{w}, {\vec{θ}}_{0}^{'})}_{L^{2}} = 0}$ it holds $\begin{array}{l} 0 & < {\overset{ˇ}{ν}}_{0} : = inf_{\vec{w} \in H^{2} {(R, C)}^{m} \cap {(ker {\overset{ˇ}{L}}_{0})}^{⊥}, ‖ \vec{w} ‖_{L^{2}} = 1} {({\overset{ˇ}{L}}_{0} \vec{w}, \vec{w})}_{L^{2} {(R, C)}^{m}} \\ = inf_{\vec{w} \in H^{1} {(R, C)}^{m} \cap {(ker {\overset{ˇ}{L}}_{0})}^{⊥}, ‖ \vec{w} ‖_{L^{2}} = 1} \int_{R} {| {\vec{w}}^{'} |}^{2} + {(D^{2} W ({\vec{θ}}_{0}) \vec{w}, \vec{w})}_{C^{m}} d z . \end{array}$
Proof.
That ${\overset{ˇ}{L}}_{0}$ is self-adjoint e.g. follows from [54, Proposition 1.1] or with a typical argument with the symmetry of ${\overset{ˇ}{L}}_{0}$ and $ρ ({\overset{ˇ}{L}}_{0}) \cap R \neq \emptyset$ by the Lax–Milgram Theorem. The property ${\overset{ˇ}{L}}_{0} ⩾ 0$ is an outcome of the energetic approach in the proof of Theorem 4.9 (first for $K = R$ , then it follows for $K = C$ ). Moreover, one can use Persson’s Theorem and Weyl sequences to show the identity for $σ_{e} ({\overset{ˇ}{L}}_{0})$ , cf. the proof of [54, Proposition 2.1]. The remaining assertions can be deduced in the analogous way as in the scalar case, cf. the proof of [5, Lemma 2.5]. □

4.3.3. The linearized ODE

We have to consider the ODE that arises from the linearization of (4.11) at ${\vec{θ}}_{0}$ . More precisely, for $\vec{A} : R \to R^{m}$ with suitable regularity and decay we seek a function $\vec{u} : R \to R^{m}$ such that $\begin{matrix} (4.14) & {\overset{ˇ}{L}}_{0} \vec{u} = \vec{A} and \overset{ˇ}{B} \vec{u} = 0, \end{matrix}$ where $\overset{ˇ}{B} \in L (H^{l} {(R)}^{m}, R)$ for some $l \in {0, 1, 2}$ with $\overset{ˇ}{B} {\vec{θ}}_{0}^{'} \neq 0$ . As before we make the assumption $dim ker {\overset{ˇ}{L}}_{0} = 1$ , cf. Remark 4.11. The problem (4.14) appears in the higher orders of the inner asymptotic expansion of (vAC), cf. Section 5.2.1 below.

The additional condition with $\overset{ˇ}{B}$ is not important, but imposed in order to get uniqueness below. The natural choice from a functional analytic point of view is $\overset{ˇ}{B} : = {(\cdot, {\vec{θ}}_{0}^{'})}_{L^{2} {(R)}^{m}} : L^{2} {(R)}^{m} \to R$ .

Theorem 4.13.
Let $dim ker {\overset{ˇ}{L}}_{0} = 1$ , cf. Remark 4.11 . Let $\overset{ˇ}{B} \in L (H^{l} {(R)}^{m}, R)$ for some $l \in {0, 1, 2}$ with $\overset{ˇ}{B} {\vec{θ}}_{0}^{'} \neq 0$ . Then it holds
Let $\vec{A} \in L^{2} {(R)}^{m}$ . Then there is a $\vec{u} \in H^{2} {(R)}^{m}$ such that ${\overset{ˇ}{L}}_{0} \vec{u} = \vec{A}$ if and only if $\int_{R} \vec{A} \cdot {\vec{θ}}_{0}^{'} = 0$ . In this case $\vec{u}$ is unique up to multiples of ${\vec{θ}}_{0}^{'}$ . In particular, ( 4.14 ) admits a unique solution $\vec{u} \in H^{2} {(R)}^{m}$ if and only if $\int_{R} \vec{A} \cdot {\vec{θ}}_{0}^{'} = 0$ . Moreover, for all $k \in N_{0}$ $\begin{matrix} {\overset{ˇ}{L}}_{0} : {\vec{u} \in H^{k + 2} {(R)}^{m} : \overset{ˇ}{B} \vec{u} = 0} \to {\vec{A} \in H^{k} {(R)}^{m} : \int_{R} \vec{A} \cdot {\vec{θ}}_{0}^{'} = 0} \end{matrix}$ is an isomorphism and the inverse is bounded by some $c (\overset{ˇ}{B}, k) > 0$ .

There is a ${\overset{ˇ}{β}}_{0} > 0$ small such that for all $β \in (0, {\overset{ˇ}{β}}_{0})$ and $k \in N_{0}$ $\begin{matrix} {\overset{ˇ}{L}}_{0} : {\vec{u} \in H_{(β)}^{k + 2} {(R)}^{m} : \overset{ˇ}{B} \vec{u} = 0} \to {\vec{A} \in H_{(β)}^{k} {(R)}^{m} : \int_{R} \vec{A} \cdot {\vec{θ}}_{0}^{'} = 0} \end{matrix}$ is an isomorphism and the norm of the inverse is bounded by a constant $C (\overset{ˇ}{B}, k) > 0$ .

Remark 4.14.

Dependence on parameteres. In Theorem 4.13 we obtained linear solution operators in suitable spaces with exponential decay. Therefore, if the right hand side in the first differential equation of (4.14) depends on additional parameters and satisfies such exponential decay estimates, the regularity and decay carries over to the solution.

Using Theorem 4.13 one can directly obtain a result for right hand sides $\vec{A}$ that converge with appropriate rate to non-zero vectors ${\vec{A}}^{\pm} \in R^{m}$ at $\pm \infty$ . The idea is as follows: set $\begin{matrix} {\vec{U}}_{\pm} : = {[D^{2} W ({\vec{u}}_{\pm})]}^{- 1} ({\vec{A}}_{\pm}) and \vec{U} : = Ξ ({\vec{U}}_{-}, {\vec{U}}_{+}), \end{matrix}$ where the latter is analogous to (4.12). Then formally it holds ${\overset{ˇ}{L}}_{0} \vec{u} = \vec{A}$ if and only if ${\overset{ˇ}{L}}_{0} (\vec{u} - \vec{U}) = \vec{A} - {\vec{A}}_{0}$ , where ${\vec{A}}_{0} : = {\overset{ˇ}{L}}_{0} \vec{U}$ . To this equation one can apply the results in Theorem 4.13 for suitable $\vec{A}$ . Note that the compatibility condition is the same as before since $\int_{R} {\vec{A}}_{0} \cdot {\vec{θ}}_{0}^{'} = 0$ due to integration by parts. The case ${\vec{A}}^{\pm} \neq 0$ is not needed here but may be interesting for more sophisticated equations, e.g. a vector-valued Cahn–Hilliard equation. Moreover, the idea to reduce to ${\vec{A}}_{\pm} = 0$ could also be helpful for triple junction cases. Finally, note the analogy in the limits at infinity to the ones in Theorem 4.2.

Kusche [54, Proposition 1.6 and Corollary 1.2] yield pointwise exponential decay estimates. The latter would be enough for our purpose. But the downside is that the exponent shrinks. Moreover, the proof of Theorem 4.13 is self-contained and simpler. However, in [54] there are also uniform estimates for finite large intervals which are important for the spectral estimates, cf. [67].

Proof of Theorem 4.13. Ad 1.
Consider $K = R$ in Lemma 4.12. With the latter one can show that $\begin{matrix} {\overset{ˇ}{L}}_{0}^{⊥} : = {\overset{ˇ}{L}}_{0} |_{{(ker {\overset{ˇ}{L}}_{0})}^{⊥}} : H^{2} {(R)}^{m} \cap {(ker {\overset{ˇ}{L}}_{0})}^{⊥} \to {(ker {\overset{ˇ}{L}}_{0})}^{⊥} \end{matrix}$ is well-defined, self-adjoint and $σ ({\overset{ˇ}{L}}_{0}) = σ ({\overset{ˇ}{L}}_{0}^{⊥}) \cup {0}$ . As in the proof of [5, Lemma 2.5] it follows that $σ ({\overset{ˇ}{L}}_{0}^{⊥}) = σ ({\overset{ˇ}{L}}_{0}) ∖ {0}$ and hence $0 \in ρ ({\overset{ˇ}{L}}_{0}^{⊥})$ . Moreover, the graph norm is equivalent to the $H^{2} {(R)}^{m}$ -norm. This can be seen via direct estimates.

Now let $\vec{v} \in H^{2} {(R)}^{m}$ solve ${\overset{ˇ}{L}}_{0} \vec{v} = \vec{A}$ for some $\vec{A} \in L^{2} {(R)}^{m}$ . Then due to integration by parts it holds ${(\vec{A}, {\vec{θ}}_{0}^{'})}_{L^{2} {(R)}^{m}} = {(\vec{v}, {\overset{ˇ}{L}}_{0} {\vec{θ}}_{0}^{'})}_{L^{2} {(R)}^{m}} = 0$ . On the other hand, let $\vec{A} \in L^{2} {(R)}^{m}$ be such that $\int_{R} \vec{A} \cdot {\vec{θ}}_{0}^{'} = 0$ . The latter is equivalent to $\vec{A} \in {(ker {\overset{ˇ}{L}}_{0})}^{⊥} \cap L^{2} {(R)}^{m}$ . The above considerations yield a unique solution $\vec{v} \in H^{2} {(R)}^{m} \cap {(ker {\overset{ˇ}{L}}_{0})}^{⊥}$ to ${\overset{ˇ}{L}}_{0}^{⊥} \vec{v} = \vec{A}$ and $‖ \vec{v} ‖_{H^{2} {(R)}^{m}} ⩽ C ‖ \vec{A} ‖_{L^{2} {(R)}^{m}}$ . Because of the assumption $dim ker {\overset{ˇ}{L}}_{0} = 1$ it holds $ker {\overset{ˇ}{L}}_{0} = span {{\vec{θ}}_{0}^{'}}$ . This implies the uniqueness in $H^{2} {(R)}^{m}$ up to multiples of ${\vec{θ}}_{0}^{'}$ . Due to $\overset{ˇ}{B} \in L (H^{l} {(R)}^{2}, R)$ for some $l \in {0, 1, 2}$ and $\overset{ˇ}{B} {\vec{θ}}_{0}^{'} \neq 0$ , we obtain that $\begin{matrix} \vec{u} : = \vec{v} - \frac{\overset{ˇ}{B} \vec{v}}{\vec{B} {\vec{θ}}_{0}^{'}} {\vec{θ}}_{0}^{'} \in H^{2} {(R)}^{m} \end{matrix}$ is well-defined, the unique solution to (4.14) and that the estimate $‖ \vec{u} ‖_{H^{2} {(R)}^{m}} ⩽ C (\overset{ˇ}{B}) ‖ \vec{A} ‖_{L^{2} {(R)}^{m}}$ holds. In particular the claim follows for $k = 0$ . For arbitrary $k \in N$ let $\vec{A} \in H^{k} {(R)}^{m}$ and $\vec{u} \in H^{2} {(R)}^{m}$ solve ${\overset{ˇ}{L}}_{0} \vec{u} = \vec{A}$ . Then it follows iteratively from the equation that $\vec{u} \in H^{k + 2} {(R)}^{m}$ . In particular ${\overset{ˇ}{L}}_{0}$ is an isomorphism with respect to the spaces in the theorem for all $k \in N$ . The estimate for the inverse can also be shown iteratively using the equation. □
Ad 2.
We prove this with similar ideas as in Section 4.2.2, i.e. for $\vec{A} \in L_{(β)}^{2} {(R)}^{m}$ such that $\int_{R} \vec{A} \cdot {\vec{θ}}_{0}^{'} = 0$ we make the ansatz $\vec{u} = e^{- β \hat{η}} \vec{v}$ with $\vec{v} \in H^{2} {(R)}^{m}$ , where $\hat{η} : R \to R$ is as in Definition 2.4, 3. It holds $\begin{matrix} {\vec{u}}^{″} = e^{- β \hat{η}} [{\vec{v}}^{″} - 2 β {\hat{η}}^{'} {\vec{v}}^{'} + \vec{v} (β^{2} {({\hat{η}}^{'})}^{2} - β {\hat{η}}^{″})] . \end{matrix}$ Therefore we consider the equation $\begin{matrix} {\overset{ˇ}{L}}_{0} \vec{v} + {\overset{ˇ}{N}}_{β} \vec{v} = e^{β \hat{η}} \vec{A}, {\overset{ˇ}{N}}_{β} \vec{v} : = 2 β {\hat{η}}^{'} {\vec{v}}^{'} - \vec{v} (β^{2} {({\hat{η}}^{'})}^{2} - β {\hat{η}}^{″}) . \end{matrix}$ In order to solve this, we want to use the spaces in 1. One problem is the compatibility condition for ${\overset{ˇ}{N}}_{β} \vec{v}$ and the right hand side. Therefore we solve a different equation, where suitable terms are subtracted that enforce the compatibility condition. Moreover, $\vec{v}$ should satisfy ${\overset{ˇ}{B}}_{(β)} \vec{v} = 0$ , where ${\overset{ˇ}{B}}_{(β)} : = \overset{ˇ}{B} (e^{- β \hat{η}} \cdot)$ . This is a problem since then the space would depend on β. Therefore note that it is enough to show the assertion e.g. for $\overset{ˇ}{B} = {\overset{ˇ}{B}}_{0} : = {(\cdot)}_{1} |_{z = 0}$ since in the end one can subtract ${\vec{θ}}_{0}^{'} \overset{ˇ}{B} \vec{u} / \overset{ˇ}{B} {\vec{θ}}_{0}^{'}$ to get a solution of (4.14). Here ${(\cdot)}_{1}$ denotes the first component. If ${\overset{ˇ}{β}}_{0} > 0$ is small enough, in particular ${\overset{ˇ}{β}}_{0} ⩽ \sqrt{λ / 2}$ with λ as in Theorem 4.9, the assertion carries over to general $\overset{ˇ}{B}$ . The advantage of ${\overset{ˇ}{B}}_{0}$ is that ${\overset{ˇ}{B}}_{0} (e^{- β \hat{η}} \vec{v}) = 0$ if and only if ${\overset{ˇ}{B}}_{0} \vec{v} = 0$ . Therefore the space for $\vec{v}$ can be chosen uniformly in β. Hence we solve ${\overset{ˇ}{B}}_{0} \vec{v} = 0$ together with $\begin{array}{l} (4.15) & {\overset{ˇ}{L}}_{0} \vec{v} + {\overset{ˇ}{M}}_{β} \vec{v} = e^{β \hat{η}} \vec{A} - [\int_{R} e^{β \hat{η}} \vec{A} \cdot {\vec{θ}}_{0}^{'}] \frac{{\vec{θ}}_{0}^{'}}{‖ {\vec{θ}}_{0}^{'} ‖_{L^{2} {(R)}^{m}}}, \\ {\overset{ˇ}{M}}_{β} \vec{v} : = {\overset{ˇ}{N}}_{β} \vec{v} - [\int_{R} {\overset{ˇ}{N}}_{β} \vec{v} \cdot {\vec{θ}}_{0}^{'}] \frac{{\vec{θ}}_{0}^{'}}{‖ {\vec{θ}}_{0}^{'} ‖_{L^{2} {(R)}^{m}}} . \end{array}$ Using the decay of ${\vec{θ}}_{0}^{'}$ from Theorem 4.3.1 and properties of exponentially weighted Sobolev spaces in Lemma 2.5 one can directly show that $\begin{matrix} {\overset{ˇ}{N}}_{β}, {\overset{ˇ}{M}}_{β} \in L (H_{(β)}^{2} {(R)}^{m}, L_{(β)}^{2} {(R)}^{m}) \end{matrix}$ with norm bounded by $C β$ for all $β \in [0, {\overset{ˇ}{β}}_{0})$ and any fixed ${\overset{ˇ}{β}}_{0} > 0$ . Therefore the first part of the theorem and a Neumann series argument imply that for all $β \in [0, {\overset{ˇ}{β}}_{0})$ and ${\overset{ˇ}{β}}_{0} > 0$ small $\begin{matrix} {\overset{ˇ}{L}}_{0} + {\overset{ˇ}{M}}_{β} : {\vec{v} \in H^{2} {(R)}^{m} : {\overset{ˇ}{B}}_{0} \vec{v} = 0} \to {\vec{A} \in L^{2} {(R)}^{m} : \int_{R} \vec{A} \cdot {\vec{θ}}_{0}^{'} = 0} \end{matrix}$ is an isomorphism and the norm of the inverse is bounded by a constant independent of β. Hence (4.15) admits a unique solution $\vec{v} \in H^{2} {(R)}^{m}$ with ${\overset{ˇ}{B}}_{0} \vec{v} = 0$ for all $β \in [0, {\overset{ˇ}{β}}_{0})$ . The computations above yield that $\vec{u} : = e^{- β \hat{η}} \vec{v} \in H_{(β)}^{2} {(R)}^{m}$ solves $\begin{matrix} {\overset{ˇ}{L}}_{0} \vec{u} = \vec{A} + [- \int_{R} e^{β \hat{η}} \vec{A} \cdot {\vec{θ}}_{0}^{'} + \int_{R} {\overset{ˇ}{N}}_{β} \vec{v} \cdot {\vec{θ}}_{0}^{'}] \frac{e^{- β \hat{η}} {\vec{θ}}_{0}^{'}}{‖ {\vec{θ}}_{0}^{'} ‖_{L^{2} {(R)}^{m}}^{2}} . \end{matrix}$ By assumption it holds $\int_{R} \vec{A} \cdot {\vec{θ}}_{0}^{'} = 0$ and because of the first part of the theorem we have $\int_{R} {\overset{ˇ}{L}}_{0} \vec{u} \cdot {\vec{θ}}_{0}^{'} = 0$ . Due to Theorem 4.9 it holds ${\vec{θ}}_{0}^{'} \neq 0$ and hence $\int_{R} e^{- β \hat{η}} | {\vec{θ}}_{0}^{'} |^{2} > 0$ . Therefore $\begin{matrix} {\overset{ˇ}{L}}_{0} \vec{u} = \vec{A} and {\overset{ˇ}{B}}_{0} \vec{u} = 0 . \end{matrix}$ Finally, the first part in the theorem yields uniqueness and the estimate follows with the above considerations and Lemma 2.5. □

4.4. A vector-valued elliptic problem on $R_{+}^{2}$ with Neumann boundary condition

This section is analogous to Section 4.2, where the scalar case was done. Let $W : R^{m} \to R$ be as in Definition 1.1 and ${\vec{θ}}_{0}$ be as in Remark 4.10, 1. In the contact point expansion for (vAC) (cf. Section 5.2.2 below) in any dimension $N ⩾ 2$ the following model problem appears: For suitable data $\vec{G} : \overline{R_{+}^{2}} \to R^{m}$ , $\vec{g} : R \to R^{m}$ find a solution $\vec{u} : \overline{R_{+}^{2}} \to R^{m}$ to the system $\begin{array}{l} (4.16) & [- Δ + D^{2} W ({\vec{θ}}_{0} (R))] \vec{u} (R, H) = \vec{G} (R, H) for (R, H) \in R_{+}^{2}, \\ (4.17) & - \partial_{H} \vec{u} |_{H = 0} (R) = \vec{g} (R) for R \in R . \end{array}$ As often in Section 4.3 we make the assumption that $dim ker {\overset{ˇ}{L}}_{0} = 1$ , where ${\overset{ˇ}{L}}_{0}$ is defined in (4.13), cf. also Remark 4.11. This implies a useful estimate for functions orthogonal to ${\vec{θ}}_{0}^{'}$ , cf. Lemma 4.12. The solution strategy for (4.16)–(4.17) is completely analogous to Section 4.2. First of all, we show some assertions for weak solutions of the problem in Section 4.4.1. Then in Section 4.4.2 we obtain solution operators in exponentially weighted Sobolev spaces.

4.4.1. Weak solutions and regularity

In the vector-valued case weak solutions are defined as follows:

Definition 4.15.
Let $\vec{G} \in L^{2} {(R_{+}^{2})}^{m}$ and $\vec{g} \in L^{2} {(R)}^{m}$ . Then $\vec{u} \in H^{1} {(R_{+}^{2})}^{m}$ is called weak solution of (4.16)–(4.17) if for all $\vec{φ} \in H^{1} {(R_{+}^{2})}^{m}$ it holds that $\begin{array}{l} \overset{ˇ}{a} (\vec{u}, \vec{φ}) & : = \int_{R_{+}^{2}} \nabla \vec{u} : \nabla \vec{φ} + {(D^{2} W ({\vec{θ}}_{0} (R)) \vec{u}, \vec{φ})}_{R^{m}} d (R, H) \\ = \int_{R_{+}^{2}} \vec{G} \cdot \vec{φ} d (R, H) + \int_{R} \vec{g} (R) \cdot \vec{φ} |_{H = 0} (R) d R . \end{array}$

We obtain the analogue of Theorem 4.4:
Theorem 4.16.
Let $dim ker {\overset{ˇ}{L}}_{0} = 1$ and consider $\vec{G} \in L^{2} {(R_{+}^{2})}^{m}$ and $\vec{g} \in L^{2} {(R)}^{m}$ . Then
$\overset{ˇ}{a} : H^{1} {(R_{+}^{2})}^{m} \times H^{1} {(R_{+}^{2})}^{m} \to R$ is not coercive.

If $\vec{G} (\cdot, H), \vec{g} ⊥ {\vec{θ}}_{0}^{'}$ for a.e. $H > 0$ in $L^{2} {(R)}^{m}$ , then there exists a weak solution $\vec{u}$ with $\vec{u} (\cdot, H) ⊥ {\vec{θ}}_{0}^{'}$ for a.e. $H > 0$ and it holds $‖ \vec{u} ‖_{H^{1} {(R_{+}^{2})}^{m}} ⩽ C (‖ \vec{G} ‖_{L^{2} {(R_{+}^{2})}^{m}} + ‖ \vec{g} ‖_{L^{2} {(R)}^{m}})$ .

Weak solutions are unique.

If $\vec{G} \cdot {\vec{θ}}_{0}^{'} \in L^{1} (R_{+}^{2})$ and $\vec{u}$ is a weak solution with $\partial_{H} \vec{u} \cdot {\vec{θ}}_{0}^{'} \in L^{1} (R_{+}^{2})$ , then the following compatibility condition holds: $\begin{matrix} (4.18) & \int_{R_{+}^{2}} \vec{G} (R, H) \cdot {\vec{θ}}_{0}^{'} (R) d (R, H) + \int_{R} \vec{g} (R) \cdot {\vec{θ}}_{0}^{'} (R) d R = 0 . \end{matrix}$

If $\vec{G} \cdot {\vec{θ}}_{0}^{'} \in L^{1} (R_{+}^{2})$ , then $\tilde{G} (H) : = {(\vec{G} (\cdot, H), {\vec{θ}}_{0}^{'})}_{L^{2} {(R)}^{m}}$ is well-defined for a.e. $H > 0$ and $\tilde{G} \in L^{1} (R_{+}) \cap L^{2} (R_{+})$ . Moreover, we have the decomposition $\begin{matrix} (4.19) & \vec{G} = \tilde{G} (H) \frac{{\vec{θ}}_{0}^{'} (R)}{‖ {\vec{θ}}_{0}^{'} ‖_{L^{2} {(R)}^{m}}^{2}} + {\vec{G}}^{⊥} (R, H), \vec{g} = {(\vec{g}, {\vec{θ}}_{0}^{'})}_{L^{2} {(R)}^{m}} \frac{{\vec{θ}}_{0}^{'} (R)}{‖ {\vec{θ}}_{0}^{'} ‖_{L^{2} {(R)}^{m}}^{2}} + {\vec{g}}^{⊥} (R), \end{matrix}$ where ${\vec{G}}^{⊥} \in L^{2} {(R_{+}^{2})}^{m}, {\vec{g}}^{⊥} \in L^{2} {(R)}^{m}$ with ${\vec{G}}^{⊥} (\cdot, H), {\vec{g}}^{⊥} ⊥ {\vec{θ}}_{0}^{'}$ in $L^{2} {(R)}^{m}$ f.a.e. $H > 0$ .

If $‖ \vec{G} (\cdot, H) ‖_{L^{2} {(R)}^{m}} ⩽ C e^{- ν H}$ for a.e. $H > 0$ and a constant $ν > 0$ , then $\vec{G} \cdot {\vec{θ}}_{0}^{'} \in L^{1} (R_{+}^{2})$ . Let $\tilde{G}$ be defined as in 5. and the compatibility condition ( 4.18 ) hold. Then $\begin{matrix} (4.20) & {\vec{u}}_{1} (R, H) : = - \int_{H}^{\infty} \int_{\tilde{H}}^{\infty} \tilde{G} (\hat{H}) d \hat{H} d \tilde{H} \frac{{\vec{θ}}_{0}^{'} (R)}{‖ {\vec{θ}}_{0}^{'} ‖_{L^{2} {(R)}^{m}}^{2}} \end{matrix}$ is well-defined for a.e. $(R, H) \in R_{+}^{2}$ , ${\vec{u}}_{1} \in W_{1}^{2} {(R_{+}^{2})}^{m} \cap H^{2} {(R_{+}^{2})}^{m}$ and ${\vec{u}}_{1}$ is a weak solution of ( 4.16 )–( 4.17 ) for $\vec{G} - {\vec{G}}^{⊥}, \vec{g} - {\vec{g}}^{⊥}$ from ( 4.19 ) instead of $\vec{G}, \vec{g}$ .

Proof.
One can essentially copy the proof of Theorem 4.4. All multiplications of functions that are now vector-valued have to be interpreted as scalar products and $f^{″} (θ_{0})$ has to be replaced by $D^{2} W ({\vec{θ}}_{0})$ . Moreover, all spaces except for products and for $\tilde{G}$ change to vector-valued ones. Finally, one uses Lemma 4.12 instead of [5, Lemma 2.5] to get coercivity of $\overset{ˇ}{a}$ on the orthogonal parts and uniqueness of weak solutions. □

Again this yields an existence theorem for weak solutions analogously to Corollary 4.5: Corollary 4.17.
Let $dim ker {\overset{ˇ}{L}}_{0} = 1$ . Then it holds
Let $\vec{g} \in L^{2} {(R)}^{m}$ , $\vec{G} \in L^{2} {(R_{+}^{2})}^{m}$ with $‖ \vec{G} (\cdot, H) ‖_{L^{2} {(R)}^{m}} ⩽ C e^{- ν H}$ for a.e. $H > 0$ and some $ν > 0$ . Let ( 4.18 ) hold. Then there is a unique weak solution of ( 4.16 )–( 4.17 ).

Let $k \in N_{0}$ and $\vec{u} \in H^{1} {(R_{+}^{2})}^{m}$ be a weak solution of ( 4.16 )–( 4.17 ) for $\vec{G} \in H^{k} {(R_{+}^{2})}^{m}$ and $\vec{g} \in H^{k + \frac{1}{2}} {(R)}^{m}$ . Then $\vec{u} \in H^{k + 2} {(R_{+}^{2})}^{m} ↪ C^{k, γ} {(\overline{R_{+}^{2}})}^{m}$ for all $γ \in (0, 1)$ and $\begin{matrix} ‖ \vec{u} ‖_{H^{k + 2} {(R_{+}^{2})}^{m}} ⩽ C_{k} (‖ \vec{G} ‖_{H^{k} {(R_{+}^{2})}^{m}} + ‖ \vec{g} ‖_{H^{k + \frac{1}{2}} {(R)}^{m}} + ‖ \vec{u} ‖_{H^{1} {(R_{+}^{2})}^{m}}) . \end{matrix}$

Proof.
The first part is a direct consequence of Theorem 4.16. The second assertion can be shown similar to the proof of [5, Corollary 2.10, 2.] using iteratively scalar regularity theory for every component of the elliptic equation, where $D^{2} W ({\vec{θ}}_{0}) \vec{u}$ is viewed as part of the right hand side. □

4.4.2. Solution operators in exponentially weighted spaces

With analogous adjustments as above one obtains solution operators in exponentially weighted Sobolev spaces similar to Section 4.2.2, cf. Theorems 4.6–4.8. We will just need the analogue of Theorem 4.8, hence we only formulate the latter in the vector-valued setting:

Theorem 4.18 (Solution Operators for the Vector-valued Case).

Let $dim ker {\overset{ˇ}{L}}_{0} = 1$ . There exist $\overset{ˇ}{γ} > 0$ and $\overset{ˇ}{β} : (0, \overset{ˇ}{γ}] \to (0, \infty)$ non-decreasing such that $\begin{array}{l} {\overset{ˇ}{L}}_{\frac{π}{2}} : = (- Δ + D^{2} W ({\vec{θ}}_{0} (R)), - \partial_{H} |_{H = 0}) : H_{(β, γ)}^{k + 2} {(R_{+}^{2})}^{m} \to {\overset{ˇ}{Y}}_{(β, γ)}^{k}, \\ {\overset{ˇ}{Y}}_{(β, γ)}^{k} : = {(\vec{G}, \vec{g}) \in H_{(β, γ)}^{k} {(R_{+}^{2})}^{m} \times H_{(β)}^{k + \frac{1}{2}} {(R)}^{m} : \int_{R_{+}^{2}} \vec{G} \cdot {\vec{θ}}_{0}^{'} + \int_{R} \vec{g} \cdot {\vec{θ}}_{0}^{'} = 0} \end{array}$ is well-defined and invertible for all $k \in N_{0}$ , $γ \in (0, \overset{ˇ}{γ}]$ and $β \in [0, \overset{ˇ}{β} (γ)]$ and the operator norm of the inverse is bounded by $\overset{ˇ}{C} (k) {(1 + \frac{1}{γ^{2}})}^{k + 1}$ .

5. Asymptotic expansions

In this section we carry out the rigorous asymptotic expansions for (AC) and (vAC) in the situations mentioned in the introduction. The expansions are based on the curvilinear coordinates from Section 3 and use the solutions for the model problems in Section 4.

5.1. Asymptotic expansion of (AC) in ND

Let $N ⩾ 2$ , $Ω \subseteq R^{N}$ be as in Remark 1.3, 1. and $Γ : = {(Γ_{t})}_{t \in [0, T]}$ be as in Section 3.1 with contact angle $\frac{π}{2}$ . We use the notation from Section 3.1 and 3.2. Moreover, let $δ > 0$ be such that the assertions of Theorem 3.2 hold for $2 δ$ instead of δ. Finally, we assume that Γ evolves according to (MCF). Moreover, let f be as in (1.1). Based on Γ we construct a smooth approximate solution $u_{ε}^{A}$ to (AC1)–(AC3) with $u_{ε}^{A} = \pm 1$ in $Q_{T}^{\pm} ∖ Γ (2 δ)$ , increasingly steep “transition” from $- 1$ to 1 for $ε \to 0$ and such that ${u_{ε}^{A} = 0}$ approaches Γ for $ε \to 0$ . This is the analogous qualitative behaviour as in the 2D-case, cf. [5, Section 3]. Compared to the latter case, the computations are similar, but more technical and we also iterate the construction. The most striking insight is that in the contact point expansion we also end up with model problems on the half space $R_{+}^{2}$ , where elements of $\partial Σ$ enter as independent variables.

Let $M \in N$ with $M ⩾ 2$ . Then we consider height functions $h_{j} : Σ \times [0, T] \to R$ for $j = 1, \dots, M$ and we set $h_{ε} : = \sum_{j = 1}^{M} ε^{j - 1} h_{j}$ . Analogously as in the 2-dimensional case we define $h_{M + 1} : = h_{M + 2} : = 0$ and we introduce the scaled variable $\begin{matrix} (5.1) & ρ_{ε} (x, t) : = \frac{r (x, t)}{ε} - h_{ε} (s (x, t), t) for (x, t) \in \overline{Γ (2 δ)} . \end{matrix}$ Note that the idea behind this definition is that ${ρ_{ε} = 0}$ should formally give the zero-level set of $u_{ε}^{A}$ which is intended to be an approximation of Γ. In formal asymptotic calculations one would start with ansatz functions depending on $\frac{r}{ε}$ , but it turns out that some refinement is needed to improve the error. The above choice of rescaled variable and the ansatz for the interior expansion below is known to work well for Allen–Cahn-type equations. There are several ansatz variants that are more suited (or necessary) for other types of equations. See [66, Section 5] for a survey of existing ansatz types.

In Section 5.1.1 we construct the inner expansion and in Section 5.1.2 the contact point expansion. Finally, in Section 5.1.3 we show that the construction yields a suitable approximate solution $u_{ε}^{A}$ to (AC1)–(AC3).

5.1.1. Inner expansion of (AC) in ND

For the inner expansion we consider the following ansatz: Let $ε > 0$ be small and $\begin{matrix} u_{ε}^{I} : = \sum_{j = 0}^{M + 1} ε^{j} u_{j}^{I}, u_{j}^{I} (x, t) : = {\hat{u}}_{j}^{I} (ρ_{ε} (x, t), s (x, t), t) for (x, t) \in \overline{Γ (2 δ)}, \end{matrix}$ where $\begin{matrix} {\hat{u}}_{j}^{I} : R \times Σ \times [0, T] \to R : (ρ, s, t) \mapsto {\hat{u}}_{j}^{I} (ρ, s, t) \end{matrix}$ for $j = 0, \dots, M + 1$ . Moreover, we set $u_{M + 2}^{I} : = 0$ and ${\hat{u}}_{ε}^{I} : = \sum_{j = 0}^{M + 1} ε^{j} {\hat{u}}_{j}^{I}$ . We will expand (AC1) for $u_{ε} = u_{ε}^{I}$ into ε-series with coefficients in $(ρ_{ε}, s, t)$ up to $O (ε^{M - 1})$ . This yields equations of analogous form as in [5, Section 3.1], where I is replaced by Σ. Therefore we have to compute the action of the differential operators on $u_{ε}^{I}$ .

In the following the surface gradient $\nabla_{Σ}$ (see [32, Definition 2.21]) for functions $g : Σ \to R$ is viewed as a map $\nabla_{Σ} g : Σ \to R^{N}$ . Then we set ${(\nabla_{Σ})}_{i} g : = {(\nabla_{Σ} g)}_{i}$ for $i = 1, \dots, N$ . If g depends on other variables as well, the analogous definition applies.

Lemma 5.1.
Let $ε > 0$ , $\hat{w} : R \times Σ \times [0, T] \to R$ be sufficiently smooth and $w : \overline{Γ (2 δ)} \to R$ be defined by $w (x, t) : = \hat{w} (ρ_{ε} (x, t), s (x, t), t)$ for all $(x, t) \in \overline{Γ (2 δ)}$ . Then $\begin{array}{l} \partial_{t} w = \partial_{ρ} \hat{w} [\frac{\partial_{t} r}{ε} - (\partial_{t} h_{ε} + \partial_{t} s \cdot \nabla_{Σ} h_{ε})] + \partial_{t} s \cdot \nabla_{Σ} \hat{w} + \partial_{t} \hat{w}, \\ \nabla w = \partial_{ρ} \hat{w} [\frac{\nabla r}{ε} - {(D_{x} s)}^{⊤} \nabla_{Σ} h_{ε}] + {(D_{x} s)}^{⊤} \nabla_{Σ} \hat{w}, \\ \begin{matrix} Δ w & = \partial_{ρ} \hat{w} [\frac{Δ r}{ε} - (Δ s \cdot \nabla_{Σ} h_{ε} + \sum_{i, l = 1}^{N} \nabla s_{i} \cdot \nabla s_{l} {(\nabla_{Σ})}_{i} {(\nabla_{Σ})}_{l} h_{ε})] \\ + Δ s \cdot \nabla_{Σ} \hat{w} + \sum_{i, l = 1}^{N} \nabla s_{i} \cdot \nabla s_{l} {(\nabla_{Σ})}_{i} {(\nabla_{Σ})}_{l} \hat{w} \\ + 2 ({(D_{x} s)}^{⊤} \nabla_{Σ} \partial_{ρ} \hat{w}) \cdot [\frac{\nabla r}{ε} - {(D_{x} s)}^{⊤} \nabla_{Σ} h_{ε}] + \partial_{ρ}^{2} \hat{w} {| \frac{\nabla r}{ε} - {(D_{x} s)}^{⊤} \nabla_{Σ} h_{ε} |}^{2}, \end{matrix} \end{array}$ where the w-terms on the left hand side and derivatives of r or s are evaluated at $(x, t) \in \overline{Γ (2 δ)}$ , the $h_{ε}$ -terms at $(s (x, t), t)$ and the $\hat{w}$ -terms at $(ρ_{ε} (x, t), s (x, t), t)$ .
Remark 5.2.

The differential operator $\nabla_{Σ}$ commutes with other ones acting on different variables. This can be shown directly with the definitions or suitable extension arguments. For the latter there are similar ideas in the proof of Lemma 5.1.

Note the similarity to [5, Lemma 3.1]. The terms on the right hand side are more sophisticated, but are always of the same type as in the 2D-case. Therefore in the expansion they will contribute in the analogous way.

Proof of Lemma 5.1.
Basically, this follows from the chain and product rules as well as from the properties of $\nabla_{Σ}$ . Let $g : Σ \to R$ be $C^{1}$ . Then $g \circ s : \overline{Γ (2 δ)} \to R$ . In order to compute $\nabla_{x} [g (s)]$ , let $\overline{g} : R^{N} \to R$ and $\overline{s} : R^{N + 1} \to R^{N}$ be smooth extensions of g and s, respectively. Such a $\overline{g}$ can be constructed via local extensions in submanifold charts and the existence of $\overline{s}$ follows from Theorem 3.2, 1. Then the chain rule yields $\begin{matrix} D_{x} (g (s)) |_{(x, t)} = D \overline{g} |_{s (x, t)} D_{x} \overline{s} |_{(x, t)} = {(\nabla_{Σ} g)}^{⊤} |_{s (x, t)} D_{x} s |_{(x, t)}, \end{matrix}$ where we have used $\nabla_{Σ} g |_{s} = P_{T_{s} Σ} \nabla \overline{g} |_{s}$ for all $s \in Σ$ due to [32, Remark 2.22] as well as $\partial_{x_{j}} s |_{(x, t)} \in T_{s (x, t)} Σ$ for all $(x, t) \in \overline{Γ (2 δ)}$ , cf. Theorem 3.2. Alternatively, one can also use the chain rule for differentials and the definition of the surface gradient. For the derivative in time this works analogously. Therefore it holds for all $(x, t) \in \overline{Γ (2 δ)}$ : $\begin{matrix} \nabla_{x} [g (s)] |_{(x, t)} = {(D_{x} s)}^{⊤} |_{(x, t)} \nabla_{Σ} g |_{s (x, t)} and \partial_{t} [g (s)] |_{(x, t)} = \partial_{t} s |_{(x, t)} \cdot \nabla_{Σ} g |_{s (x, t)} . \end{matrix}$ With similar arguments and the chain rule one can derive formulas for the first derivatives of functions of type $g (s (x, t), t)$ for $(x, t) \in \overline{Γ (2 δ)}$ , where $g : Σ \times [0, T] \to R$ . In this case it holds $\begin{array}{l} \frac{d}{d t} [g (s (x, \cdot), \cdot)] |_{t} = \partial_{t} g |_{(s (x, t), t)} + \partial_{t} s |_{(x, t)} \cdot \nabla_{Σ} g |_{(s (x, t), t)}, \\ \nabla_{x} [g (s (\cdot, t), t)] |_{x} = (D_{x} s) |_{(x, t)}^{⊤} \nabla_{Σ} g |_{(s (x, t), t)} . \end{array}$ Using this and similar arguments as before, one can derive formulas for the first derivatives of functions of the form $g (ρ_{ε} (x, t), s (x, t), t)$ for $(x, t) \in \overline{Γ (2 δ)}$ with $g : R \times Σ \times [0, T] \to R$ . The latter are written in the lemma for $\hat{w}$ instead of g. Putting all those identities together and using the product rule in $\partial_{x_{j}} {(\nabla w)}_{j}$ for $j = 1, \dots, N$ , one obtains the formula for $Δ w$ . □

To expand the Allen–Cahn equation $\partial_{t} u_{ε}^{I} - Δ u_{ε}^{I} + \frac{1}{ε^{2}} f^{'} (u_{ε}^{I}) = 0$ into ε-series, we again use Taylor expansions. For the $f^{'}$ -part this is identically to the 2D-case: If the $u_{j}^{I}$ are bounded, then $\begin{matrix} (5.2) & f^{'} (u_{ε}^{I}) = f^{'} (u_{0}^{I}) + \sum_{k = 1}^{M + 2} \frac{f^{(k + 1)} (u_{0}^{I})}{k!} {[\sum_{j = 1}^{M + 1} u_{j}^{I} ε^{j}]}^{k} + O (ε^{M + 3}) on \overline{Γ (2 δ)} . \end{matrix}$ The terms in the ε-expansion that are needed explicitly are $\begin{array}{l} O (1) : f^{'} (u_{0}^{I}), \\ O (ε) : f^{″} (u_{0}^{I}) u_{1}^{I}, \\ O (ε^{2}) : f^{″} (u_{0}^{I}) u_{2}^{I} + \frac{f^{(3)} (u_{0}^{I})}{2!} {(u_{1}^{I})}^{2} . \end{array}$ For $k = 3, \dots, M + 1$ the order $O (ε^{k})$ is given by $\begin{array}{l} O (ε^{k}) : f^{″} (u_{0}^{I}) u_{k}^{I} + [ & some polynomial in (u_{1}^{I}, \dots, u_{k - 1}^{I}) of order ⩽ k, where the \\ coefficients are multiples of f^{(3)} (u_{0}^{I}), \dots, f^{(k + 1)} (u_{0}^{I}) \\ and every term contains a u_{j}^{I} -factor] . \end{array}$ Let $u_{M + 2}^{I} : = 0$ . Then the latter also holds for $k = M + 2$ . The other explicit terms in (5.2) are of order $O (ε^{M + 3})$ .

Moreover, we expand functions of $(x, t) \in \overline{Γ (2 δ)}$ into ε-series with a Taylor expansion via $r (x, t) = ε (ρ_{ε} (x, t) + h_{ε} (s (x, t), t))$ for $(x, t) \in \overline{Γ (2 δ)}$ . Then again $ρ_{ε}$ is replaced by $ρ \in R$ . For a smooth $g : \overline{Γ (2 δ)} \to R$ the Taylor expansion yields for $r \in [- 2 δ, 2 δ]$ uniformly in $(s, t)$ : $\begin{matrix} (5.3) & \tilde{g} (r, s, t) : = g (\overline{X} (r, s, t)) = \sum_{k = 0}^{M + 2} \frac{\partial_{r}^{k} \tilde{g} |_{(0, s, t)}}{k!} r^{k} + O (| r |^{M + 3}) . \end{matrix}$ Only the first few terms in the ε-expansion are needed explicitly. These are $\begin{array}{l} O (1) : g |_{{\overline{X}}_{0} (s, t)}, \\ O (ε) : (ρ + h_{1} (s, t)) \partial_{r} \tilde{g} |_{(0, s, t)}, \\ O (ε^{2}) : h_{2} (s, t) \partial_{r} \tilde{g} |_{(0, s, t)} + {(ρ + h_{1} (s, t))}^{2} \frac{\partial_{r}^{2} \tilde{g} |_{(0, s, t)}}{2} . \end{array}$ For $k = 3, \dots, M$ the order $O (ε^{k})$ is $\begin{array}{l} O (ε^{k}) : & h_{k} \partial_{r} \tilde{g} |_{(0, s, t)} + \frac{\partial_{r}^{2} \tilde{g} |_{(0, s, t)}}{2} 2 (ρ + h_{1} (s, t)) h_{k - 1} (s, t) \\ + [some polynomial in (ρ, h_{1} (s, t), \dots, h_{k - 2} (s, t)) of order ⩽ k, \\ where the coefficients are multiples of \partial_{r}^{2} \tilde{g} |_{(0, s, t)}, \dots, \partial_{r}^{k} \tilde{g} |_{(0, s, t)}] . \end{array}$ Let $h_{M + 1} : = h_{M + 2} : = 0$ . Then the latter also holds for $k = M + 1, M + 2$ . The other explicit terms in (5.3) are bounded by $ε^{M + 3}$ times some polynomial in $| ρ |$ if the $h_{j}$ are bounded. Later, these terms and the $O (| r |^{M + 3})$ -term in (5.3) for each choice of g will be multiplied with terms that decay exponentially in $| ρ |$ . Then these remainder terms will become $O (ε^{M + 3})$ .

For the higher orders in the expansion the following definition is useful:
Definition 5.3 (Notation for Inner Expansion of (AC) in ND).

We call $(θ_{0}, u_{1}^{I})$ the zero-th inner order and $(h_{j}, u_{j + 1}^{I})$ the jth inner order for $j = 1, \dots, M$ .

Let $k \in {- 1, \dots, M + 2}$ . We denote with $P_{k}^{I}$ the set of polynomials in ρ with smooth coefficients in $(s, t) \in Σ \times [0, T]$ depending only on the $h_{j}$ for $1 ⩽ j ⩽ min {k, M}$ .

Let $k \in {- 1, \dots, M + 2}$ and $β > 0$ . We denote with $R_{k, (β)}^{I}$ the set of smooth functions $R : R \times Σ \times [0, T] \to R$ that depend only on the jth inner orders for $0 ⩽ j ⩽ min {k, M}$ and satisfy uniformly in $(ρ, s, t)$ : $\begin{matrix} | \partial_{ρ}^{i} {(\nabla_{Σ})}_{n_{1}} \dots {(\nabla_{Σ})}_{n_{d}} \partial_{t}^{n} R (ρ, s, t) | = O (e^{- β | ρ |}) \end{matrix}$ for all $n_{1}, \dots, n_{d} \in {1, \dots, N}$ and $d, i, n \in N_{0}$ .

For $k \in {- 1, \dots, M + 2}$ and $β > 0$ the set ${\hat{R}}_{k, (β)}^{I}$ is defined analogously to $R_{k, (β)}^{I}$ with functions of type $R : R \times \partial Σ \times [0, T] \to R$ instead.4

⁴
Note that this set only appears in the contact point expansion later.

Now we expand (AC1) for $u_{ε} = u_{ε}^{I}$ into ε-series. This works analogously to the 2D-case, cf. Remark 5.2, 2. In the following $(ρ, s, t)$ are always in $R \times Σ \times [0, T]$ and sometimes omitted.

Inner expansion: $O (ε^{- 2})$ . We obtain that the $O (\frac{1}{ε^{2}})$ -order is zero if $\begin{matrix} - | \nabla r |^{2} |_{{\overline{X}}_{0} (s, t)} \partial_{ρ}^{2} {\hat{u}}_{0}^{I} (ρ, s, t) + f^{'} ({\hat{u}}_{0}^{I} (ρ, s, t)) = 0 . \end{matrix}$ Because of Theorem 3.2 we have $| \nabla r |^{2} |_{{\overline{X}}_{0} (s, t)} = 1$ . Moreover, ${ρ_{ε} = 0}$ should approximate the zero level set of $u_{ε}^{I}$ . Hence we require ${\hat{u}}_{0}^{I} (0, s, t) = 0$ . Finally, ${\hat{u}}_{0}^{I}$ should connect the values $\pm 1$ , i.e. ${lim}_{ρ \to \pm \infty} {\hat{u}}_{0}^{I} (ρ, s, t) = \pm 1$ . Altogether due to Theorem 4.1 we have to define $\begin{matrix} {\hat{u}}_{0}^{I} (ρ, s, t) : = θ_{0} (ρ) . \end{matrix}$

Inner expansion: $O (ε^{- 1})$ . The $\partial_{t} u$ -part yields $\frac{1}{ε} \partial_{t} r |_{{\overline{X}}_{0} (s, t)} θ_{0}^{'} (ρ)$ and $Δ u$ yields $\begin{array}{l} \frac{1}{ε^{2}} [\partial_{r} (| \nabla r |^{2} \circ \overline{X}) |_{(0, s, t)} ε (ρ + h_{1} (s, t)) θ_{0}^{″} (ρ) + | \nabla r |^{2} |_{{\overline{X}}_{0} (s, t)} ε \partial_{ρ}^{2} {\hat{u}}_{1}^{I} (ρ, s, t)] \\ + \frac{1}{ε} [θ_{0}^{'} (ρ) Δ r |_{{\overline{X}}_{0} (s, t)} + 2 {(D_{x} s \nabla r)}^{⊤} |_{{\overline{X}}_{0} (s, t)} (\nabla_{Σ} θ_{0}^{'} (ρ) - \nabla_{Σ} h_{1} (s, t) θ_{0}^{″} (ρ))] \\ = \frac{1}{ε} [\partial_{ρ}^{2} {\hat{u}}_{1}^{I} (ρ, s, t) + θ_{0}^{'} (ρ) Δ r |_{{\overline{X}}_{0} (s, t)}], \end{array}$ where we used Theorem 3.2. Therefore the $O (\frac{1}{ε})$ -order cancels if $\begin{matrix} L_{0} {\hat{u}}_{1}^{I} (ρ, s, t) + θ_{0}^{'} (ρ) (\partial_{t} r - Δ r) |_{{\overline{X}}_{0} (s, t)} = 0, where L_{0} : = - \partial_{ρ}^{2} + f^{″} (θ_{0}) . \end{matrix}$ Due to Theorem 4.2 this parameter-dependent ODE together with ${\hat{u}}_{1}^{I} (0, s, t) = 0$ and boundedness in ρ has a (unique) solution ${\hat{u}}_{1}^{I}$ if and only if $(\partial_{t} r - Δ r) |_{{\overline{X}}_{0} (s, t)} = 0$ . The latter is valid because it is equivalent to (MCF) for Γ by Theorem 3.2. Therefore we define ${\hat{u}}_{1}^{I} : = 0$ .

Inner expansion: $O (ε^{0})$ . From $\partial_{t} u$ we obtain $\begin{array}{l} \frac{1}{ε} [\partial_{t} r |_{{\overline{X}}_{0} (s, t)} ε \partial_{ρ} {\hat{u}}_{1}^{I} + \partial_{r} (\partial_{t} r \circ \overline{X}) |_{(0, s, t)} ε (ρ + h_{1} (s, t)) θ_{0}^{'} (ρ)] \\ + θ_{0}^{'} (ρ) [- \partial_{t} h_{1} (s, t) - \partial_{t} s |_{{\overline{X}}_{0} (s, t)} \cdot \nabla_{Σ} h_{1} (s, t)] + \partial_{t} s |_{{\overline{X}}_{0} (s, t)} \cdot \nabla_{Σ} θ_{0} + \partial_{t} θ_{0} (ρ) \\ = θ_{0}^{'} (ρ) [(ρ + h_{1} (s, t)) \partial_{r} (\partial_{t} r \circ \overline{X}) |_{(0, s, t)} - \partial_{t} h_{1} (s, t) - \partial_{t} s |_{{\overline{X}}_{0} (s, t)} \cdot \nabla_{Σ} h_{1} (s, t)], \end{array}$ and from $Δ u$ : $\begin{array}{l} \frac{1}{ε^{2}} θ_{0}^{″} (ρ) [ε^{2} \frac{1}{2} {(ρ + h_{1})}^{2} \partial_{r}^{2} (| \nabla r |^{2} \circ \overline{X}) |_{(0, s, t)} + ε^{2} h_{2} \partial_{r} (| \nabla r |^{2} \circ \overline{X}) |_{(0, s, t)}] \\ + \frac{1}{ε^{2}} \partial_{ρ}^{2} {\hat{u}}_{1}^{I} ε^{2} (ρ + h_{1}) \partial_{r} (| \nabla r |^{2} \circ \overline{X}) |_{(0, s, t)} + \frac{1}{ε^{2}} | \nabla r |^{2} |_{{\overline{X}}_{0} (s, t)} ε^{2} \partial_{ρ}^{2} {\hat{u}}_{2}^{I} \\ + \frac{1}{ε} [θ_{0}^{'} (ρ) ε (ρ + h_{1}) \partial_{r} (Δ r \circ \overline{X}) |_{(0, s, t)} + ε \partial_{ρ} {\hat{u}}_{1}^{I} Δ r |_{{\overline{X}}_{0} (s, t)}] \\ + \frac{1}{ε} 2 {(D_{x} s \nabla r)}^{⊤} |_{{\overline{X}}_{0} (s, t)} [\nabla_{Σ} \partial_{ρ} {\hat{u}}_{1}^{I} ε - \nabla_{Σ} h_{1} ε \partial_{ρ}^{2} {\hat{u}}_{1}^{I} - ε \nabla_{Σ} h_{2} θ_{0}^{″} (ρ)] \\ + \frac{1}{ε} 2 \partial_{r} ({(D_{x} s \nabla r)}^{⊤} \circ \overline{X}) |_{(0, s, t)} ε (ρ + h_{1}) [\nabla_{Σ} θ_{0}^{'} (ρ) - \nabla_{Σ} h_{1} θ_{0}^{″} (ρ)] \\ + Δ s |_{{\overline{X}}_{0} (s, t)} \cdot \nabla_{Σ} θ_{0} (ρ) + \sum_{i, l = 1}^{N} \nabla s_{i} \cdot \nabla s_{l} |_{{\overline{X}}_{0} (s, t)} {(\nabla_{Σ})}_{i} {(\nabla_{Σ})}_{l} θ_{0} (ρ) \\ - 2 \nabla_{Σ} θ_{0}^{'} {(ρ)}^{⊤} D_{x} s {(D_{x} s)}^{⊤} |_{{\overline{X}}_{0} (s, t)} \nabla_{Σ} h_{1} + | {(D_{x} s)}^{⊤} |_{{\overline{X}}_{0} (s, t)} \nabla_{Σ} h_{1} |^{2} θ_{0}^{″} (ρ) \\ - θ_{0}^{'} (ρ) [Δ s |_{{\overline{X}}_{0} (s, t)} \cdot \nabla_{Σ} h_{1} + \sum_{i, l = 1}^{N} \nabla s_{i} \cdot \nabla s_{l} |_{{\overline{X}}_{0} (s, t)} {(\nabla_{Σ})}_{i} {(\nabla_{Σ})}_{l} h_{1}] . \end{array}$ Because of Theorem 3.2 the latter is the same as $\begin{array}{l} θ_{0}^{″} (ρ) [\frac{1}{2} {(ρ + h_{1})}^{2} \partial_{r}^{2} (| \nabla r |^{2} \circ \overline{X}) |_{(0, s, t)} + {| {(D_{x} s)}^{⊤} |_{{\overline{X}}_{0} (s, t)} \nabla_{Σ} h_{1} |}^{2}] + \partial_{ρ}^{2} {\hat{u}}_{2}^{I} \\ + θ_{0}^{″} (ρ) 2 \partial_{r} ({(D_{x} s \nabla r)}^{⊤} \circ \overline{X}) |_{(0, s, t)} (ρ + h_{1}) (- \nabla_{Σ} h_{1}) + θ_{0}^{'} (ρ) (ρ + h_{1}) \partial_{r} (Δ r \circ \overline{X}) |_{(0, s, t)} \\ - θ_{0}^{'} (ρ) [Δ s |_{{\overline{X}}_{0} (s, t)} \cdot \nabla_{Σ} h_{1} + \sum_{i, l = 1}^{N} \nabla s_{i} \cdot \nabla s_{l} {(\nabla_{Σ})}_{i} {(\nabla_{Σ})}_{l} h_{1}] . \end{array}$ Due to ${\hat{u}}_{1}^{I} = 0$ , the $f^{'}$ -part contributes $f^{″} (θ_{0}) {\hat{u}}_{2}^{I}$ . Hence for the cancellation of the $O (1)$ -term in the expansion for the Allen–Cahn equation (AC1) for $u_{ε} = u_{ε}^{I}$ we require $\begin{array}{l} (5.4) & - L_{0} {\hat{u}}_{2}^{I} (ρ, s, t) = R_{1} (ρ, s, t), \\ \begin{matrix} R_{1} (ρ, s, t) : = & θ_{0}^{'} (ρ) [- \partial_{t} h_{1} + \sum_{i, l = 1}^{N} \nabla s_{i} \cdot \nabla s_{l} {(\nabla_{Σ})}_{i} {(\nabla_{Σ})}_{l} h_{1} \\ + (ρ + h_{1}) \partial_{r} ((\partial_{t} r - Δ r) \circ \overline{X}) |_{(0, s, t)} - (\partial_{t} s - Δ s) |_{{\overline{X}}_{0} (s, t)} \cdot \nabla_{Σ} h_{1}] \\ + θ_{0}^{″} (ρ) [- \frac{1}{2} {(ρ + h_{1})}^{2} \partial_{r}^{2} (| \nabla r |^{2} \circ \overline{X}) |_{(0, s, t)} \\ + 2 (ρ + h_{1}) \partial_{r} ({(D_{x} s \nabla r)}^{⊤} \circ \overline{X}) |_{(0, s, t)} \nabla_{Σ} h_{1} - {| {(D_{x} s)}^{⊤} |_{{\overline{X}}_{0} (s, t)} \nabla_{Σ} h_{1} |}^{2}] . \end{matrix} \end{array}$ If $h_{1}$ is smooth, then $R_{1}$ is smooth and together with all derivatives decays exponentially in $| ρ |$ uniformly in $(s, t)$ with rate β for every $β \in (0, min {\sqrt{f^{″} (\pm 1)}})$ because of Theorem 4.1. Therefore Theorem 4.2 applied in local coordinates for Σ yields that there is a unique bounded solution ${\hat{u}}_{2}^{I}$ to (5.4) together with ${\hat{u}}_{2}^{I} (0, s, t) = 0$ if and only if the compatibility condition $\int_{R} R_{1} (ρ, s, t) θ_{0}^{'} (ρ) d ρ = 0$ holds. Since $\int_{R} θ_{0}^{'} (ρ) θ_{0}^{″} (ρ) d ρ = 0$ due to integration by parts, the nonlinearities in $h_{1}$ drop out and we obtain a linear non-autonomous parabolic equation for $h_{1}$ on Σ with principal part $\partial_{t} - \sum_{i, l = 1}^{N} \nabla s_{i} \cdot \nabla s_{l} |_{{\overline{X}}_{0} (s, t)} {(\nabla_{Σ})}_{i} {(\nabla_{Σ})}_{l}$ : $\begin{matrix} (5.5) & \partial_{t} h_{1} - \sum_{i, l = 1}^{N} \nabla s_{i} \cdot \nabla s_{l} |_{{\overline{X}}_{0} (s, t)} {(\nabla_{Σ})}_{i} {(\nabla_{Σ})}_{l} h_{1} + a_{1} \cdot \nabla_{Σ} h_{1} + a_{0} h_{1} = f_{0} \end{matrix}$ in $Σ \times [0, T]$ . Here, with $d_{1}, \dots, d_{5}$ defined by $\begin{array}{l} d_{1} : = \int_{R} θ_{0}^{'} {(ρ)}^{2} d ρ, d_{2} : = \int_{R} θ_{0}^{'} {(ρ)}^{2} ρ d ρ, d_{3} : = \int_{R} θ_{0}^{'} {(ρ)}^{2} ρ^{2} d ρ, \\ d_{4} : = \int_{R} θ_{0}^{'} (ρ) θ_{0}^{″} (ρ) ρ d ρ, d_{5} : = \int_{R} θ_{0}^{'} (ρ) θ_{0}^{″} (ρ) ρ^{2} d ρ, d_{6} : = \int_{R} θ_{0}^{'} (ρ) θ_{0}^{″} (ρ) ρ^{3} d ρ, \end{array}$ we have set for all $(s, t) \in Σ \times [0, T]$ : $\begin{array}{l} (5.6) & a_{1} (s, t) : = (\partial_{t} s - Δ s) |_{{\overline{X}}_{0} (s, t)} - 2 \frac{d_{4}}{d_{1}} \partial_{r} ({(D_{x} s \nabla r)}^{⊤} \circ \overline{X}) |_{(0, s, t)} \in R^{N}, \\ (5.7) & a_{0} (s, t) : = - \partial_{r} ((\partial_{t} r - Δ r) \circ \overline{X}) |_{(0, s, t)} + \frac{d_{4}}{d_{1}} \partial_{r}^{2} (| \nabla r |^{2} \circ \overline{X}) |_{(0, s, t)} \in R, \\ (5.8) & f_{0} (s, t) : = \frac{d_{2}}{d_{1}} \partial_{r} ((\partial_{t} r - Δ r) \circ \overline{X}) |_{(0, s, t)} - \frac{d_{5}}{2 d_{1}} \partial_{r}^{2} (| \nabla r |^{2} \circ \overline{X}) |_{(0, s, t)} \in R . \end{array}$ If $h_{1}$ is smooth and solves (5.5), then Theorem 4.2 (applied in local coordinates for Σ) yields a smooth solution ${\hat{u}}_{2}^{I}$ to (5.4) and we also get decay estimates. By compactness, we obtain ${\hat{u}}_{2}^{I} \in R_{1, (β)}^{I}$ for any $β \in (0, min {\sqrt{f^{″} (\pm 1)}})$ because of the following remark:

Remark 5.4.

The norm of the entirety of derivatives in local coordinates up to any fixed order $d \in N$ is equivalent to the norm of the collection of all $\nabla_{Σ}$ -derivatives up to order d on any compact subset of a chart domain. This can be shown inductively via local representations.

Remark 5.5.

If additionally f is even, then $θ_{0}^{'}$ is even and $θ_{0}^{″}$ is odd. Hence $d_{2} = d_{5} = 0$ and $f_{0} = 0$ . Therefore the equation (5.5) for $h_{1}$ is homogeneous in this case.

Inner expansion: $O (ε^{k})$ . Let $k \in {1, \dots, M - 1}$ and suppose that the jth inner order has already been constructed for $j = 0, \dots, k$ , that it is smooth and ${\hat{u}}_{j + 1}^{I} \in R_{j, (β)}^{I}$ for every $β \in (0, min {\sqrt{f^{″} (\pm 1)}})$ . This assumption will be fulfilled later, when we apply an induction argument.

Then with the notation in Definition 5.3 it holds for all $β \in (0, min {\sqrt{f^{″} (\pm 1)}})$ : $\begin{array}{l} For j = 1, \dots, k + 2 : [O (ε^{j}) in (5.2)] \in f^{″} (u_{0}^{I}) u_{j}^{I} + R_{j - 2, (β)}^{I} [\subseteq R_{j - 1, (β)}^{I}, if j ⩽ k + 1], \\ For j = 1, \dots, k + 1 : [O (ε^{j}) in (5.3)] \in \partial_{r} \tilde{g} |_{(0, s, t)} h_{j} + P_{j - 1}^{I} [\subseteq P_{j}^{I}, if j ⩽ k], \\ For j = 3, \dots, k + 1 : [O (ε^{j}) in (5.3)] \in \partial_{r} \tilde{g} |_{(0, s, t)} h_{j} + \partial_{r}^{2} \tilde{g} |_{(0, s, t)} (ρ + h_{1}) h_{j - 1} + P_{j - 2}^{I} . \end{array}$

With these identities one can compute the $O (ε^{k})$ -order in (AC1) for $u_{ε} = u_{ε}^{I}$ . The calculation is straightforward but lengthy, cf. [66, Section 5.1.1.4] for the 2D-case. Therefore we do not go into details. However, note that the explicit identities from Theorem 3.2 enter as well as that Γ evolves according to (MCF) and ${\hat{u}}_{1}^{I} = 0$ . This yields that the $O (ε^{k})$ -order in (AC1) for $u_{ε} = u_{ε}^{I}$ vanishes if $\begin{array}{l} (5.9) & - L_{0} {\hat{u}}_{k + 2}^{I} (ρ, s, t) = R_{k + 1} (ρ, s, t), \\ R_{k + 1} (ρ, s, t) : = θ_{0}^{'} (ρ) [- \partial_{t} h_{k + 1} + \sum_{i, l = 1}^{N} \nabla s_{i} \cdot \nabla s_{l} |_{{\overline{X}}_{0} (s, t)} {(\nabla_{Σ})}_{i} {(\nabla_{Σ})}_{l} h_{k + 1} \\ - (\partial_{t} s - Δ s) |_{{\overline{X}}_{0} (s, t)} \cdot \nabla_{Σ} h_{k + 1} + h_{k + 1} \partial_{r} ((\partial_{t} r - Δ r) \circ \overline{X}) |_{(0, s, t)}] \\ + θ_{0}^{″} (ρ) [- (ρ + h_{1}) h_{k + 1} \partial_{r}^{2} (| \nabla r |^{2} \circ \overline{X}) |_{(0, s, t)} \\ - 2 {(\nabla_{Σ} h_{1})}^{⊤} D_{x} s {(D_{x} s)}^{⊤} |_{{\overline{X}}_{0} (s, t)} \nabla_{Σ} h_{k + 1} \\ + 2 \partial_{r} ({(D_{x} s \nabla r)}^{⊤} \circ \overline{X}) |_{(0, s, t)} [(ρ + h_{1}) \nabla_{Σ} h_{k + 1} + h_{k + 1} \nabla_{Σ} h_{1}]] \\ + {\tilde{R}}_{k} (ρ, s, t), \end{array}$ where ${\tilde{R}}_{k} \in R_{k, (β)}^{I}$ . If $h_{k + 1}$ is smooth, then due to Theorem 4.2 (applied in local coordinates for Σ) equation (5.9) has a unique bounded solution ${\hat{u}}_{k + 2}^{I}$ with ${\hat{u}}_{k + 2}^{I} (0, s, t) = 0$ if and only if $\int_{R} R_{k + 1} (ρ, s, t) θ_{0}^{'} (ρ) d ρ = 0$ . Because of $\int_{R} θ_{0}^{″} θ_{0}^{'} = 0$ the latter is equivalent to $\begin{matrix} (5.10) & \partial_{t} h_{k + 1} - \sum_{i, l = 1}^{N} \nabla s_{i} \cdot \nabla s_{l} |_{{\overline{X}}_{0} (s, t)} {(\nabla_{Σ})}_{i} {(\nabla_{Σ})}_{l} h_{k + 1} + a_{1} \cdot \nabla_{Σ} h_{k + 1} + a_{0} h_{k + 1} = f_{k}, \end{matrix}$ where $\begin{matrix} f_{k} (s, t) : = \int_{R} {\tilde{R}}_{k} (ρ, s, t) θ_{0}^{'} (ρ) d ρ \frac{1}{‖ θ_{0}^{'} ‖_{L^{2} (R)}^{2}} \end{matrix}$ is a smooth function of $(s, t)$ and depends only on the jth inner orders for $0 ⩽ j ⩽ k$ . Here $a_{0}, a_{1}$ are defined in (5.6)–(5.7). If $h_{k + 1}$ is smooth and solves (5.10), then Theorem 4.2 yields as in the last subparagraph a smooth solution ${\hat{u}}_{k + 2}^{I}$ to (5.9) such that ${\hat{u}}_{k + 2}^{I} \in R_{k + 1, (β)}^{I}$ for all $β \in (0, min {\sqrt{f^{″} (\pm 1)}})$ .

5.1.2. Contact point expansion of (AC) in ND

In the contact point expansion we proceed similarly as in the 2D-case, cf. [5, Section 3.2], but the computations are more technical. Here we have more contact points, namely all $X_{0} (σ, t)$ , where $(σ, t) \in \partial Σ \times [0, T]$ . We make the ansatz $u_{ε} = u_{ε}^{I} + u_{ε}^{C}$ in $Γ (2 δ)$ close to $X_{0} (\partial Σ \times [0, T])$ . Therefore we use the mappings $Y : \partial Σ \times [0, 2 μ_{1}] \to R (Y) \subset Σ$ as well as $\begin{matrix} (σ, b) = Y^{- 1} \circ s : \overline{Γ^{C} (2 δ, 2 μ_{1})} \to \partial Σ \times [0, 2 μ_{1}] \end{matrix}$ from Section 3.2, where the set $Γ^{C} (2 δ, 2 μ_{1})$ was defined in Remark 3.3. Besides r we only scale b with ε. Let $H_{ε} : = \frac{b}{ε}$ and $\begin{matrix} u_{ε}^{C} : = \sum_{j = 1}^{M} ε^{j} u_{j}^{C}, u_{j}^{C} (x, t) : = {\hat{u}}_{j}^{C} (ρ_{ε} (x, t), H_{ε} (x, t), σ (x, t), t) for (x, t) \in \overline{Γ^{C} (2 δ, 2 μ_{1})}, \end{matrix}$ where $\begin{matrix} {\hat{u}}_{j}^{C} : \overline{R_{+}^{2}} \times \partial Σ \times [0, T] \to R : (ρ, H, σ, t) \mapsto {\hat{u}}_{j}^{C} (ρ, H, σ, t) \end{matrix}$ for $j = 1, \dots, M$ . Moreover, we set $u_{M + 1}^{C} : = u_{M + 2}^{C} : = 0$ and ${\hat{u}}_{ε}^{C} : = \sum_{j = 1}^{M} ε^{j} {\hat{u}}_{j}^{C}$ . Instead of (AC1) for $u_{ε} = u_{ε}^{I} + u_{ε}^{C}$ , we will expand $\begin{array}{l} (5.11) & \partial_{t} u_{ε}^{C} - Δ u_{ε}^{C} + \frac{1}{ε^{2}} [f^{'} (u_{ε}^{I} + u_{ε}^{C}) - f^{'} (u_{ε}^{I})] = 0 \end{array}$ into ε-series with coefficients in $(ρ_{ε}, H_{ε}, σ, t)$ . This makes sense because $u_{ε}^{I}$ should solve the equation (AC1) to sufficiently high order later and we can formally subtract the latter now. We call (5.11) the “bulk equation” and expand it up to $O (ε^{M - 2})$ . Moreover, we will expand (AC2) for $u_{ε} = u_{ε}^{I} + u_{ε}^{C}$ into ε-series with coefficients in $(ρ_{ε}, σ, t)$ up to $O (ε^{M - 1})$ . Altogether we end up with similar equations as in [5, Section 3.2]. Here besides $t \in [0, T]$ also points on $\partial Σ$ enter as independent variables in the model problems on $R_{+}^{2}$ . The solvability condition (4.5) yields the boundary conditions on $\partial Σ \times [0, T]$ for the height functions $h_{j}$ .

For the expansion we calculate the action of the differential operators on $u_{ε}^{C}$ in the next lemma. Here for $\nabla_{\partial Σ}$ and $\nabla_{Σ}$ we use similar conventions as in Lemma 5.1.

Lemma 5.6.
Let $\overline{R_{+}^{2}} \times \partial Σ \times [0, T] ∋ (ρ, H, σ, t) \mapsto \hat{w} (ρ, H, σ, t) \in R$ be sufficiently smooth and let $w : \overline{Γ^{C} (2 δ, 2 μ_{1})} \to R : (x, t) \mapsto \hat{w} (ρ_{ε} (x, t), H_{ε} (x, t), σ (x, t), t)$ . Then $\begin{array}{l} \partial_{t} w = \partial_{ρ} \hat{w} [\frac{\partial_{t} r}{ε} - (\partial_{t} h_{ε} + \partial_{t} s \cdot \nabla_{Σ} h_{ε})] + \partial_{H} \hat{w} \frac{\partial_{t} b}{ε} + \partial_{t} σ \cdot \nabla_{\partial Σ} \hat{w} + \partial_{t} \hat{w}, \\ \nabla w = \partial_{ρ} \hat{w} [\frac{\nabla r}{ε} - {(D_{x} s)}^{⊤} \nabla_{Σ} h_{ε}] + \partial_{H} \hat{w} \frac{\nabla b}{ε} + {(D_{x} σ)}^{⊤} \nabla_{\partial Σ} \hat{w}, \\ \begin{matrix} Δ w & = \partial_{ρ} \hat{w} [\frac{Δ r}{ε} - (Δ s \cdot \nabla_{Σ} h_{ε} + \sum_{i, l = 1}^{N} \nabla s_{i} \cdot \nabla s_{l} {(\nabla_{Σ})}_{i} {(\nabla_{Σ})}_{l} h_{ε})] + \partial_{H} \hat{w} \frac{Δ b}{ε} \\ + \partial_{H}^{2} \hat{w} \frac{| \nabla b |^{2}}{ε^{2}} + \partial_{ρ}^{2} \hat{w} {| \frac{\nabla r}{ε} - {(D_{x} s)}^{⊤} \nabla_{Σ} h_{ε} |}^{2} + 2 \partial_{ρ} \partial_{H} \hat{w} \frac{\nabla b}{ε} \cdot [\frac{\nabla r}{ε} - {(D_{x} s)}^{⊤} \nabla_{Σ} h_{ε}] \\ + 2 ({(D_{x} σ)}^{⊤} \nabla_{\partial Σ} \partial_{ρ} \hat{w}) \cdot [\frac{\nabla r}{ε} - {(D_{x} s)}^{⊤} \nabla_{Σ} h_{ε}] + 2 ({(D_{x} σ)}^{⊤} \nabla_{\partial Σ} \partial_{H} \hat{w}) \cdot \frac{\nabla b}{ε} \\ + Δ σ \cdot \nabla_{\partial Σ} \hat{w} + \sum_{i, l = 1}^{N} \nabla σ_{i} \cdot \nabla σ_{l} {(\nabla_{\partial Σ})}_{i} {(\nabla_{\partial Σ})}_{l} \hat{w}, \end{matrix} \end{array}$ where the w-terms on the left hand side and derivatives of r or s are evaluated at $(x, t)$ , the $h_{ε}$ -terms at $(s (x, t), t)$ and the $\hat{w}$ -terms at $(ρ_{ε} (x, t), H_{ε} (x, t), σ (x, t), t)$ .
Proof.
This can be shown in a similar manner as in the proof of Lemma 5.1. □
Remark 5.7.
The formulas in Lemma 5.6 without the $\nabla_{\partial Σ}$ -terms directly correspond to [5, Lemma 3.3]. The structure of the new terms is similar whereas their ε-order is the same or higher. Hence later these will only contribute to lower order remainder terms in the expansion.

Contact point expansion: The bulk equation. We expand the $f^{'}$ -part in (5.11): If the $u_{j}^{I}, u_{j}^{C}$ are bounded, the Taylor expansion yields on $\overline{Γ (2 δ)}$ $\begin{matrix} (5.12) & f^{'} (u_{ε}^{I} + u_{ε}^{C}) = f^{'} (θ_{0}) + \sum_{k = 1}^{M + 2} \frac{1}{k!} f^{(k + 1)} (θ_{0}) {[\sum_{j = 1}^{M + 1} ε^{j} (u_{j}^{I} + u_{j}^{C})]}^{k} + O (ε^{M + 3}) . \end{matrix}$ Combining this with the expansion for $f^{'} (u_{ε}^{I})$ in (5.2) and using $u_{1}^{I} = 0$ , the terms in the asymptotic expansion for $f^{'} (u_{ε}^{I} + u_{ε}^{C}) - f^{'} (u_{ε}^{I})$ are for $k = 1, \dots, M + 1$ : $\begin{array}{l} O (1) : 0, \\ O (ε) : f^{″} (θ_{0}) u_{1}^{C}, \\ \begin{matrix} O (ε^{k}) : f^{″} (θ_{0}) u_{k}^{C} + [ & some polynomial in (u_{1}^{I}, \dots, u_{k - 1}^{I}, u_{1}^{C}, \dots, u_{k - 1}^{C}) of order ⩽ k, \\ where the coefficients are multiples of f^{(3)} (θ_{0}), \dots, f^{(k + 1)} (θ_{0}) \\ and every term contains a u_{j}^{C} -factor] . \end{matrix} \end{array}$ Let $u_{M + 2}^{C} : = 0$ . Then the latter is also valid for $k = M + 2$ . The other explicit terms in $f^{'} (u_{ε}^{I} + u_{ε}^{C}) - f^{'} (u_{ε}^{I})$ are of order $O (ε^{M + 3})$ .

Moreover, we expand terms arising from Lemma 5.6 in (5.11) that are functions of $(s, t)$ or $(ρ, s, t)$ , i.e. all the $h_{j}$ -terms and the $u_{j}^{I}$ -terms from the $f^{'}$ -expansion, respectively, as well as the terms depending on $(x, t)$ , i.e. all the derivatives of $r, b, s, σ$ . Therefore we consider smooth $g_{1} : Σ \times [0, T] \to R$ or $g_{1} : R \times Σ \times [0, T] \to R$ such that ${\tilde{g}}_{1} : = g_{1} |_{s = Y}$ admits bounded derivatives in b, where $Y : \partial Σ \times [0, 2 μ_{1}] \to Σ : (σ, b) \mapsto Y (σ, b)$ . Due to $b = ε H_{ε}$ we apply a Taylor expansion to obtain uniformly $\begin{matrix} (5.13) & g_{1} |_{b = ε H} = {\tilde{g}}_{1} |_{b = 0} + \sum_{k = 1}^{M + 2} {(ε H)}^{k} \frac{\partial_{b}^{k} {\tilde{g}}_{1} |_{b = 0}}{k!} + O ({(ε H)}^{M + 3}) for H \in [0, \frac{2 μ_{1}}{ε}] . \end{matrix}$ Furthermore, let $g_{2} : \overline{Γ^{C} (2 δ, 2 μ_{1})} \to R$ be smooth. For convenience we define $\begin{matrix} {\overline{X}}_{1} : [- 2 δ, 2 δ] \times [0, 2 μ_{1}] \times \partial Σ \times [0, T] \to \overline{Γ^{C} (2 δ, 2 μ_{1})} : (r, b, σ, t) \mapsto \overline{X} (r, Y (σ, b), t) . \end{matrix}$ Then a Taylor expansion yields $\begin{matrix} (5.14) & {\tilde{g}}_{2} (r, b, σ, t) : = g_{2} ({\overline{X}}_{1} (r, b, σ, t)) = \sum_{j + k = 0}^{M + 2} \frac{\partial_{r}^{j} \partial_{s}^{k} {\tilde{g}}_{2} |_{(0, σ, t)}}{j! k!} r^{j} b^{k} + O ({| (r, b) |}^{M + 3}) \end{matrix}$ uniformly in $(r, b, σ, t) \in [- 2 δ, 2 δ] \times [0, 2 μ_{1}] \times \partial Σ \times [0, T]$ . Later we evaluate at $\begin{matrix} r = ε (ρ_{ε} (x, t) + h_{ε} (s (x, t), t)), b = ε H_{ε} (x, t), σ = σ (x, t) for (x, t) \in \overline{Γ^{C} (2 δ, 2 μ_{1})} \end{matrix}$ and expand $h_{ε}$ with (5.13). Then we replace $(ρ_{ε}, H_{ε})$ by arbitrary $(ρ, H) \in \overline{R_{+}^{2}}$ . Hence for $k = 1, \dots, M + 2$ we obtain $\begin{array}{l} 2 O (1) : g_{2} |_{{\overline{X}}_{0} (σ, t)} \\ O (ε) : \partial_{r} {\tilde{g}}_{2} |_{(0, σ, t)} (ρ + h_{1} |_{(σ, t)}) + \partial_{b} {\tilde{g}}_{2} |_{(0, σ, t)} H . \\ \begin{matrix} O (ε^{k}) : [ & some polynomial in (ρ, H, \partial_{b}^{l} {\tilde{h}}_{j} |_{(σ, t)}), l = 0, \dots, k - 1, j = 1, \dots, k of order ⩽ k, \\ where the coefficients are multiples of \partial_{r}^{l_{1}} \partial_{b}^{l_{2}} {\tilde{g}}_{2} |_{(0, σ, t)}, l_{1}, l_{2} \in N_{0}, l_{1} + l_{2} ⩽ k], \end{matrix} \end{array}$ where $h_{M + 1} = h_{M + 2} = 0$ by definition. The order $O (ε)$ is not needed explicitly and just included for clarity. The other explicit terms in (5.13) are estimated by $ε^{M + 3}$ times some polynomial in $(| ρ_{ε} |, H_{ε})$ . In the end these terms and the remainder in (5.13) are multiplied with exponentially decaying terms and hence become $O (ε^{M + 3})$ .

Let us introduce some notation for the higher orders in the expansion:
Definition 5.8 (Notation for Contact Point Expansion of (AC) in ND).

We denote with $(θ_{0}, u_{1}^{I})$ the zero-th order and with $(h_{j}, u_{j + 1}^{I}, u_{j}^{C})$ the jth order for $j = 1, \dots, M$ .

Let $k \in {- 1, \dots, M + 2}$ . Let $P_{k}^{C} (ρ, H)$ be the set of polynomials in $(ρ, H)$ with smooth coefficients in $(σ, t) \in \partial Σ \times [0, T]$ that depend only on the $h_{j}$ for $1 ⩽ j ⩽ min {k, M}$ . The sets $P_{k}^{C} (ρ)$ and $P_{k}^{C} (H)$ are defined with $(ρ, H)$ replaced by ρ and H, respectively.

Let $k \in {- 1, \dots, M + 2}$ and $β, γ > 0$ . Then $R_{k, (β, γ)}^{C}$ denotes the set of smooth functions $R : \overline{R_{+}^{2}} \times \partial Σ \times [0, T] \to R$ depending only on the jth orders for $0 ⩽ j ⩽ min {k, M}$ and such that uniformly in $(ρ, H, σ, t)$ : $\begin{matrix} | \partial_{ρ}^{i} \partial_{H}^{l} {(\nabla_{\partial Σ})}_{n_{1}} \dots {(\nabla_{\partial Σ})}_{n_{d}} \partial_{t}^{n} R (ρ, H, σ, t) | = O (e^{- (β | ρ | + γ H)}) \end{matrix}$ for all $n_{1}, \dots, n_{d} \in {1, \dots, N}$ and $d, i, l, n \in N_{0}$ .

The set $R_{k, (β)}^{C}$ is defined analogously to $R_{k, (β, γ)}^{C}$ but without the H-dependence.

In the following we expand the bulk equation (5.11) with the above formulas into ε-series with coefficients in $(ρ_{ε}, H_{ε}, σ, t)$ .

Bulk Equation: $O (ε^{- 1})$ . The lowest order $O (\frac{1}{ε})$ in (5.11) vanishes if $\begin{matrix} (5.15) & [- Δ (σ, t) + f^{″} (θ_{0} (ρ))] {\hat{u}}_{1}^{C} (ρ, H, σ, t) = 0, \end{matrix}$ where $Δ (σ, t) : = \partial_{ρ}^{2} + | \nabla b |^{2} |_{{\overline{X}}_{0} (σ, t)} \partial_{H}^{2}$ and we have used $\nabla r \cdot \nabla b |_{{\overline{X}}_{0} (σ, t)} = 0$ for every $(σ, t) \in \partial Σ \times [0, T]$ .

Bulk Equation: $O (ε^{k - 1})$ . For $k = 1, \dots, M - 1$ we compute $O (ε^{k - 1})$ in (5.11) and derive an equation for ${\hat{u}}_{k + 1}^{C}$ . Therefore we assume that the jth order is already constructed for $j = 0, \dots, k$ , that it is smooth and it holds ${\hat{u}}_{j + 1}^{I} \in R_{j, (β_{1})}^{I}$ for every $β_{1} \in (0, min {\sqrt{f^{″} (\pm 1)}})$ (bounded and all derivatives bounded is enough here) and we assume ${\hat{u}}_{j}^{C} \in R_{j, (β, γ)}^{C}$ for every $β \in (0, min {\overline{β} (γ), \sqrt{f^{″} (\pm 1)}})$ , $γ \in (0, \overline{γ})$ , where $\overline{β}, \overline{γ}$ are as in Theorem 4.8.

Then with the notation from Definition 5.8 it holds for all those $(β, γ)$ : $\begin{array}{l} For j = 1, \dots, k + 1 : [O (ε^{j}) in (5.12) minus (5.2)] \in f^{″} (θ_{0} (ρ)) {\hat{u}}_{j}^{C} + R_{j - 1, (β, γ)}^{C}, \\ For i, j = 1, \dots, k : [O (ε^{j}) in (5.13) for g_{1} = g_{1} (h_{i})] \in P_{i}^{C} (H), \\ For j = 0, \dots, k : [O (ε^{j}) in (5.14)] \in P_{j}^{C} (ρ, H) . \end{array}$

With these identities, one can compute that the $O (ε^{k - 1})$ -order in the expansion for the bulk equation (5.11) is zero if $\begin{matrix} (5.16) & [- Δ (σ, t) + f^{″} (θ_{0})] {\hat{u}}_{k + 1}^{C} = G_{k} (ρ, H, σ, t), \end{matrix}$ where $G_{k} \in R_{k, (β, γ)}^{C}$ . See [66, Section 5.1.2.1.2] for the computation in the 2D-case.

Contact point expansion: The Neumann boundary condition. The boundary conditions complementing (5.15) and (5.16) will be obtained from the expansion of the Neumann boundary condition (AC2) for $u_{ε} = u_{ε}^{I} + u_{ε}^{C}$ , i.e. $N_{\partial Ω} \cdot \nabla (u_{ε}^{I} + u_{ε}^{C}) |_{\partial Q_{T}} = 0$ . Lemma 5.1 and Lemma 5.6 yield on $\overline{Γ^{C} (2 δ, 2 μ_{1})}$ $\begin{array}{l} \nabla u_{ε}^{I} |_{(x, t)} = \partial_{ρ} {\hat{u}}_{ε}^{I} |_{(ρ, s, t)} [\frac{\nabla r |_{(x, t)}}{ε} - {(D_{x} s)}^{⊤} |_{(x, t)} \nabla_{Σ} h_{ε} |_{(s, t)}] + {(D_{x} s)}^{⊤} |_{(x, t)} \nabla_{Σ} {\hat{u}}_{ε}^{I} |_{(ρ, s, t)}, \\ \begin{matrix} \nabla u_{ε}^{C} |_{(x, t)} & = \partial_{ρ} {\hat{u}}_{ε}^{C} |_{(ρ, H, σ, t)} [\frac{\nabla r |_{(x, t)}}{ε} - {(D_{x} s)}^{⊤} |_{(x, t)} \nabla_{Σ} h_{ε} |_{(s, t)}] + \frac{\nabla b |_{(x, t)}}{ε} \partial_{H} {\hat{u}}_{ε}^{C} |_{(ρ, H, σ, t)} \\ + {(D_{x} σ)}^{⊤} |_{(x, t)} \nabla_{\partial Σ} {\hat{u}}_{ε}^{C} |_{(ρ, H, σ, t)}, \end{matrix} \end{array}$ where $ρ = ρ_{ε} (x, t), H = H_{ε} (x, t)$ , $s = s (x, t)$ and $σ = σ (x, t)$ . We consider the points $x = X (r, σ, t)$ for $(r, σ, t) \in [- 2 δ, 2 δ] \times \partial Σ \times [0, T]$ , in particular $H = 0$ and $s = σ$ .

For $g : \overline{Γ (2 δ)} \cap \partial Q_{T} \to R$ smooth we use a Taylor expansion similar to (5.3): $\begin{matrix} (5.17) & \tilde{g} (r, σ, t) : = g (\overline{X} (r, σ, t)) = \sum_{k = 0}^{M + 2} \frac{\partial_{r}^{k} \tilde{g} |_{(0, σ, t)}}{k!} r^{k} + O (| r |^{M + 3}) . \end{matrix}$ Then we insert $r = ε (ρ_{ε} + h_{ε} |_{(σ, t)})$ and replace $ρ_{ε}$ by an arbitrary $ρ \in R$ . Analogous to the inner expansion, the terms in the ε-expansion of (5.17) are for $k = 2, \dots, M$ : $\begin{array}{l} O (1) : g |_{{\overline{X}}_{0} (σ, t)}, \\ O (ε) : (ρ + h_{1} |_{(σ, t)}) \partial_{r} \tilde{g} |_{(0, σ, t)}, \\ \begin{matrix} O (ε^{k}) : h_{k} |_{(σ, t)} \partial_{r} \tilde{g} |_{(0, σ, t)} + [ & a polynomial in (ρ, h_{1} |_{(σ, t)}, \dots, h_{k - 1} |_{(σ, t)}) of order ⩽ k, \\ where coefficients are multiples of (\partial_{r}^{2} \tilde{g}, \dots, \partial_{r}^{k} \tilde{g}) |_{(0, σ, t)}] . \end{matrix} \end{array}$ The latter also holds for $k = M + 1, M + 2$ since $h_{M + 1} = h_{M + 2} = 0$ by definition. The other explicit terms in (5.17) are bounded by $ε^{M + 3}$ times some polynomial in $| ρ |$ if the $h_{j}$ are bounded. Later, these terms and the $O (| r |^{M + 3})$ -term in (5.17) for each choice of g will be multiplied with terms that decay exponentially in $| ρ |$ . Then these remainder terms will become $O (ε^{M + 3})$ .

In the following we expand the Neumann boundary condition into ε-series with coefficients in $(ρ_{ε}, σ, t)$ up to the order $O (ε^{M - 1})$ .

Neumann Boundary Condition: $O (ε^{- 1})$ . For the lowest order $O (\frac{1}{ε})$ we obtain the equation $(N_{\partial Ω} \cdot \nabla r) |_{{\overline{X}}_{0} (σ, t)} θ_{0}^{'} (ρ) = 0$ . This is valid due to the 90°-contact angle condition.

Neumann Boundary Condition: $O (ε^{0})$ . The order $O (1)$ is zero if we require $\begin{array}{l} (5.18) & (N_{\partial Ω} \cdot \nabla b) |_{{\overline{X}}_{0} (σ, t)} \partial_{H} {\hat{u}}_{1}^{C} |_{H = 0} (ρ, σ, t) = g_{1} (ρ, σ, t), \\ g_{1} |_{(ρ, σ, t)} : = θ_{0}^{'} [{(D_{x} s N_{\partial Ω})}^{⊤} |_{{\overline{X}}_{0} (σ, t)} \nabla_{Σ} h_{1} |_{(σ, t)} - \partial_{r} ((N_{\partial Ω} \cdot \nabla r) \circ \overline{X}) |_{(0, σ, t)} h_{1} |_{(σ, t)}] + {\tilde{g}}_{0} |_{(ρ, σ, t)}, \end{array}$ where ${\tilde{g}}_{0} (ρ, σ, t) : = - ρ θ_{0}^{'} (ρ) \partial_{r} ((N_{\partial Ω} \cdot \nabla r) \circ \overline{X}) |_{(0, σ, t)}$ . For $j = 1, \dots, M$ let $\begin{matrix} (5.19) & {\overline{u}}_{j}^{C} : \overline{R_{+}^{2}} \times \partial Σ \times [0, T] \to R : (ρ, H, σ, t) \mapsto {\hat{u}}_{j}^{C} (ρ, | \nabla b | ({\overline{X}}_{0} (σ, t)) H, σ, t) . \end{matrix}$ Note that $| \nabla b |_{{\overline{X}}_{0} (σ, t)} | ⩾ c > 0$ and $| N_{\partial Ω} \cdot \nabla b |_{{\overline{X}}_{0} (σ, t)} | ⩾ c > 0$ for all $(σ, t) \in \partial Σ \times [0, T]$ because of Theorem 3.2. Hence equations (5.15) and (5.18) for ${\hat{u}}_{1}^{C}$ are equivalent to $\begin{array}{l} (5.20) & [- Δ + f^{″} (θ_{0} (ρ))] {\overline{u}}_{1}^{C} = 0, \\ (5.21) & - \partial_{H} {\overline{u}}_{1}^{C} |_{H = 0} = (| \nabla b | / N_{\partial Ω} \cdot \nabla b) |_{{\overline{X}}_{0} (σ, t)} g_{1} (ρ, σ, t) . \end{array}$ The solvability condition (4.5) belonging to (5.20)–(5.21) is $\begin{matrix} (| \nabla b | / N_{\partial Ω} \cdot \nabla b) |_{{\overline{X}}_{0} (σ, t)} \int_{R} g_{1} (ρ, σ, t) θ_{0}^{'} (ρ) d ρ = 0 . \end{matrix}$ This yields a linear boundary condition for $h_{1}$ : $\begin{matrix} (5.22) & b_{1} (σ, t) \cdot \nabla_{Σ} h_{1} |_{(σ, t)} + b_{0} (σ, t) h_{1} |_{(σ, t)} = f_{0} (σ, t) for (σ, t) \in \partial Σ \times [0, T], \end{matrix}$ where $\begin{array}{l} b_{1} (σ, t) : = (| \nabla b | / N_{\partial Ω} \cdot \nabla b) |_{{\overline{X}}_{0} (σ, t)} (D_{x} s N_{\partial Ω}) |_{{\overline{X}}_{0} (σ, t)} \in R^{N}, \\ b_{0} (σ, t) : = - (| \nabla b | / N_{\partial Ω} \cdot \nabla b) |_{{\overline{X}}_{0} (σ, t)} \partial_{r} ((N_{\partial Ω} \cdot \nabla r) \circ \overline{X}) |_{(0, σ, t)} \in R, \\ f_{0} (σ, t) : = (| \nabla b | / N_{\partial Ω} \cdot \nabla b) |_{{\overline{X}}_{0} (σ, t)} \int_{R} θ_{0}^{'} (ρ) {\tilde{g}}_{0} (ρ, σ, t) d ρ / {‖ θ_{0}^{'} ‖}_{L^{2} (R)}^{2} \in R \end{array}$ are smooth in $(σ, t) \in \partial Σ \times [0, T]$ . Together with the linear parabolic equation (5.5) for $h_{1}$ from Section 5.1.1, we obtain a time-dependent linear parabolic boundary value problem for $h_{1}$ , where the initial value $h_{1} |_{t = 0}$ is not prescribed yet.

Remark 5.9.
If f is even, then so is $θ_{0}^{'}$ and thus $f_{0} = 0$ . Therefore the boundary condition (5.22) for $h_{1}$ is homogeneous and because of Remark 5.5 we can choose $h_{1} = 0$ in this case.

Now we solve (5.5) together with (5.22). We show that the principal part in (5.5) satisfies a suitable ellipticity condition and that (5.22) fulfils a non-tangentiality condition in local coordinates. Based on this one can show maximal regularity results in Hölder spaces (similar to Lunardi, Sinestrari, von Wahl [61]) and Sobolev spaces (similar to Prüss, Simonett [70, Chapters 6.1–6.4] and Denk, Hieber, Prüss [31]) with typical localization procedures. This always involves compatibility conditions for the initial value. In our case these can be avoided via extension arguments similar to [5, Section 3.2.2]. All these arguments involve many technical computations, but are in principle well-known. Therefore we refrain from going into details.

The ellipticity condition. Let $y : U \subseteq Σ \to V \subseteq \overline{R_{+}^{N - 1}}$ be a chart. Moreover, we denote with ${(g_{i j})}_{i, j = 1}^{N - 1} : V \to R^{(N - 1) \times (N - 1)}$ the local representation of the Euclidean metric on Σ and let ${(g^{k l})}_{k, l = 1}^{N - 1}$ denote its pointwise inverse. Then one can show with local representations that for $h : U \to R$ sufficiently smooth and $i, j = 1, \dots, N$ it holds $\begin{array}{l} [{(\nabla_{Σ})}_{i} {(\nabla_{Σ})}_{j} h] \circ y^{- 1} & = \sum_{p, q, k, l = 1}^{N - 1} g^{p q} g^{k l} \partial_{v_{q}} {(y^{- 1})}_{i} \partial_{v_{l}} {(y^{- 1})}_{j} \partial_{v_{p}} \partial_{v_{k}} (h \circ y^{- 1}) \\ + g^{p q} \partial_{v_{q}} {(y^{- 1})}_{i} [\partial_{v_{p}} g^{k l} \partial_{v_{l}} {(y^{- 1})}_{j} + g^{k l} \partial_{v_{p}} \partial_{v_{k}} {(y^{- 1})}_{j}] \partial_{v_{k}} (h \circ y^{- 1}) . \end{array}$ Therefore the principal part of $\sum_{i, j = 1}^{N} \nabla s_{i} \cdot \nabla s_{j} |_{{\overline{X}}_{0} (\cdot, t)} {(\nabla_{Σ})}_{i} {(\nabla_{Σ})}_{j}$ for fixed $t \in [0, T]$ in the local coordinates with respect to y is given by $\begin{matrix} \sum_{p, k = 1}^{N - 1} A_{p, k} \partial_{v_{p}} \partial_{v_{k}}, A_{p, k} : = \sum_{i, j = 1}^{N} \nabla s_{i} \cdot \nabla s_{j} |_{{\overline{X}}_{0} (y^{- 1}, t)} \sum_{q, l = 1}^{N - 1} g^{p q} g^{k l} \partial_{v_{q}} {(y^{- 1})}_{i} \partial_{v_{l}} {(y^{- 1})}_{j} . \end{matrix}$ For any appropriate ellipticity notion, it is enough to prove that $A : = {(A_{p, k})}_{p, k = 1}^{N - 1}$ is uniformly positive definite on compact subsets of V. First we show that A is pointwise positive definite. Therefore we represent $\partial_{x_{n}} s |_{{\overline{X}}_{0} (y^{- 1} (\cdot), t)} = \sum_{μ = 1}^{N - 1} {\tilde{A}}_{n, μ} |_{(\cdot, t)} \partial_{v_{μ}} (y^{- 1})$ on V for $n = 1, \dots, N$ and we denote $\tilde{A} : = {({\tilde{A}}_{n, μ} |_{(\cdot, t)})}_{n, μ = 1}^{N, N - 1} : V \to R^{N \times (N - 1)}$ . Then it holds for all $p, k = 1, \dots, N - 1$ $\begin{array}{l} A_{p, k} & = \sum_{i, j, n = 1}^{N} \sum_{μ, ν, q, l = 1}^{N - 1} {\tilde{A}}_{n, μ} {\tilde{A}}_{n, ν} g^{p q} g^{k l} \partial_{v_{μ}} {(y^{- 1})}_{i} \partial_{v_{q}} {(y^{- 1})}_{i} \partial_{v_{ν}} {(y^{- 1})}_{j} \partial_{v_{l}} {(y^{- 1})}_{j} \\ = \sum_{n = 1}^{N} \sum_{μ, ν, q, l = 1}^{N - 1} {\tilde{A}}_{n, μ} {\tilde{A}}_{n, ν} g^{p q} g^{k l} g_{μ q} g_{ν l} = \sum_{n = 1}^{N} \sum_{μ, ν = 1}^{N - 1} {\tilde{A}}_{n, μ} {\tilde{A}}_{n, ν} δ_{μ}^{p} δ_{ν}^{k} = {({\tilde{A}}^{⊤} \tilde{A})}_{p, k} . \end{array}$ Moreover, Theorem 3.2 yields that ${(\partial_{x_{n}} s |_{{\overline{X}}_{0} (s, t)})}_{n = 1}^{N}$ generate $T_{s} Σ$ for all $s \in Σ$ . Hence the matrix $\tilde{A} |_{v} \in R^{N \times (N - 1)}$ is injective for all $v \in V$ . Finally, this yields $\begin{matrix} w^{⊤} A w = | \tilde{A} w |^{2} > 0 for all w \in R^{N - 1} . \end{matrix}$ Therefore A is pointwise positive definite. Since it is equivalent to prove the estimate for vectors on the sphere in $R^{N - 1}$ , by compactness A is uniform positive definite on compact subsets of V. Altogether, the principal part in (5.5) satisfies a suitable ellipticity condition.

The non-tangentiality condition. Let $y : U \subseteq Σ \to V \subseteq \overline{R_{+}^{N - 1}}$ be a chart with $U \cap \partial Σ \neq \emptyset$ . Then for $h : U \to R$ sufficiently smooth and fixed $t \in [0, T]$ it holds: $\begin{matrix} b_{1} (y^{- 1}, t) \cdot (\nabla_{Σ} h \circ y^{- 1}) = \nabla_{v} (h \circ y^{- 1}) \cdot {[\sum_{l = 1}^{N - 1} g^{k l} b_{1} (y^{- 1}, t) \cdot \partial_{v_{l}} (y^{- 1})]}_{k = 1}^{N - 1} \end{matrix}$ on $V \cap (R^{N - 2} \times {0})$ , where $b_{1}$ is defined below (5.22). Therefore the transformed boundary condition in the local coordinates with respect to y satisfies a non-tangentiality condition if $\sum_{l = 1}^{N - 1} g^{N - 1, l} b_{1} (y^{- 1}, t) \cdot \partial_{v_{l}} (y^{- 1}) \neq 0$ on the set $V \cap (R^{N - 2} \times {0})$ . On the latter set we use the representation $b_{1} (y^{- 1}, t) = \sum_{n = 1}^{N - 1} B_{n} (y^{- 1}, t) \partial_{v_{n}} (y^{- 1})$ . Then the condition reads as $B_{N - 1} (σ, t) \neq 0$ for all $(σ, t) \in (U \cap \partial Σ) \times [0, T]$ . However, Theorem 3.2 yields $b_{1} (σ, t) \cdot {\vec{n}}_{\partial Σ} (σ) \neq 0$ for all $(σ, t) \in \partial Σ \times [0, T]$ . Due to the properties of y, we know that ${(\partial_{v_{l}} (y^{- 1}) |_{y (σ)})}_{l = 1}^{N - 1}$ form a basis of $T_{σ} Σ$ and the first $N - 2$ components are a basis of $T_{σ} \partial Σ$ for all $σ \in U \cap \partial Σ$ . Because ${\vec{n}}_{\partial Σ}$ is orthogonal to $T_{σ} \partial Σ$ , it necessarily holds $B_{N - 1} (σ, t) \neq 0$ for all $(σ, t) \in (U \cap \partial Σ) \times [0, T]$ . Therefore the boundary condition (5.22) satisfies a non-tangentiality condition in local coordinates.

Finally, we obtain a smooth solution $h_{1}$ to (5.5) and (5.22). Therefore ${\hat{u}}_{2}^{I}$ (solving (5.4)) is determined from Section 5.1.1 and it holds ${\hat{u}}_{2}^{I} \in R_{1, (β_{1})}^{I}$ for every $β_{1} \in (0, min {\sqrt{f^{″} (\pm 1)}})$ . In particular the first inner order is computed. Moreover, it holds $g_{1} \in {\hat{R}}_{1, (β_{1})}^{I}$ for all $β_{1}$ as above because of Theorem 4.1. Hence with Theorem 4.8 (applied in local coordinates for $\partial Σ$ ) there is a unique smooth solution ${\overline{u}}_{1}^{C}$ to (5.20)–(5.21) and we get decay properties. By compactness and Remark 5.4 with $\partial Σ$ instead of Σ we obtain the decay property ${\overline{u}}_{1}^{C} \in R_{1, (β, γ)}^{C}$ for all $β \in (0, min {\overline{β} (γ), \sqrt{f^{″} (\pm 1)}})$ , $γ \in (0, \overline{γ})$ , where $\overline{β}, \overline{γ}$ are as in Theorem 4.8. Altogether the first order is determined.

Neumann Boundary Condition: $O (ε^{k})$ and Induction. For $k = 1, \dots, M - 1$ we consider $O (ε^{k})$ in (AC2) for $u_{ε} = u_{ε}^{I} + u_{ε}^{C}$ and derive equations for the ( $k + 1$ )-th order. We assume the following induction hypothesis: suppose that the jth order already has been constructed for all $j = 0, \dots, k$ , that it is smooth and admits the decay ${\hat{u}}_{j + 1}^{I} \in R_{j, (β_{1})}^{I}$ for all $β_{1} \in (0, min {\sqrt{f^{″} (\pm 1)}})$ as well as ${\hat{u}}_{j}^{C} \in R_{j, (β, γ)}^{C}$ for every $β \in (0, min {\overline{β} (γ), \sqrt{f^{″} (\pm 1)}})$ , $γ \in (0, \overline{γ})$ , where $\overline{β}, \overline{γ}$ are as in Theorem 4.8. The assumption holds for $k = 1$ by the last subparagraph.

With the notation as in Definition 5.8 it holds for $j = 1, \dots, k + 1$ : $\begin{matrix} [O (ε^{j}) in (5.17)] \in \partial_{r} \tilde{g} |_{(0, σ, t)} h_{j} |_{(σ, t)} + P_{j - 1}^{C} (ρ) [\subseteq P_{j}^{C} (ρ), if j ⩽ k] . \end{matrix}$ With this identity one can show (see [66, Section 5.1.2.2.3] for the calculation in the 2D-case) that the $O (ε^{k})$ -order in (AC2) for $u_{ε} = u_{ε}^{I} + u_{ε}^{C}$ vanishes if $\begin{array}{l} (5.23) & (N_{\partial Ω} \cdot \nabla b) |_{{\overline{X}}_{0} (σ, t)} \partial_{H} {\hat{u}}_{k + 1}^{C} |_{H = 0} (ρ, σ, t) = g_{k + 1} (ρ, σ, t), \\ \begin{matrix} g_{k + 1} |_{(ρ, σ, t)} : = & θ_{0}^{'} (ρ) [{(D_{x} s N_{\partial Ω})}^{⊤} |_{{\overline{X}}_{0} (σ, t)} \nabla_{Σ} h_{k + 1} |_{(σ, t)} - \partial_{r} ((N_{\partial Ω} \cdot \nabla r) \circ \overline{X}) |_{(0, σ, t)} h_{k + 1} |_{(σ, t)}] \\ + {\tilde{g}}_{k} (ρ, σ, t), \end{matrix} \end{array}$ where ${\tilde{g}}_{k} \in R_{k, (β)}^{C}$ and therefore $g_{k + 1} \in {\hat{R}}_{k + 1, (β)}^{I} + R_{k, (β)}^{C}$ , if $h_{k + 1}$ is smooth.

As in the last subparagraph, the equations (5.16), (5.23) are equivalent to $\begin{array}{l} (5.24) & [- Δ + f^{″} (θ_{0} (ρ))] {\overline{u}}_{k + 1}^{C} = {\overline{G}}_{k} (ρ, H, σ, t), \\ (5.25) & - \partial_{H} {\overline{u}}_{k + 1}^{C} |_{H = 0} = (| \nabla b | / N_{\partial Ω} \cdot \nabla b) |_{{\overline{X}}_{0} (σ, t)} g_{k + 1} (ρ, H, σ, t), \end{array}$ where ${\overline{u}}_{k + 1}^{C}$ was defined in (5.19) and ${\overline{G}}_{k}$ is defined analogously with the $G_{k} \in R_{k, (β, γ)}^{C}$ from (5.16). The corresponding compatibility condition (4.5), namely $\begin{matrix} \int_{R_{+}^{2}} {\overline{G}}_{k} (ρ, H, σ, t) θ_{0}^{'} (ρ) d (ρ, H) + (| \nabla b | / N_{\partial Ω} \cdot \nabla b) |_{{\overline{X}}_{0} (σ, t)} \int_{R} g_{k + 1} (ρ, σ, t) θ_{0}^{'} (ρ) d ρ = 0, \end{matrix}$ yields a linear boundary condition for $h_{k + 1}$ : $\begin{matrix} (5.26) & b_{1} (σ, t) \cdot \nabla_{Σ} h_{k + 1} |_{(σ, t)} + b_{0} (σ, t) h_{k + 1} |_{(σ, t)} = f_{k}^{B} (σ, t) for (σ, t) \in \partial Σ \times [0, T], \end{matrix}$ where $b_{0}, b_{1}$ are defined below (5.22) and $\begin{matrix} f_{k}^{B} (σ, t) : = - \frac{1}{‖ θ_{0}^{'} ‖_{L^{2} (R)}^{2}} [\int_{R_{+}^{2}} {\overline{G}}_{k} |_{(ρ, H, σ, t)} θ_{0}^{'} (ρ) d ρ + \frac{| \nabla b |}{N_{\partial Ω} \cdot \nabla b} |_{{\overline{X}}_{0} (σ, t)} \int_{R} {\tilde{g}}_{k} |_{(ρ, σ, t)} θ_{0}^{'} (ρ) d ρ] \end{matrix}$ is smooth in $(σ, t) \in \partial Σ \times [0, T]$ .

Because of the computations in the last subparagraph one can solve (5.10) from Section 5.1.1 together with (5.26) and get a smooth solution $h_{k + 1}$ . Therefore Section 5.1.1 yields ${\hat{u}}_{k + 2}^{I}$ (solving (5.9)) with ${\hat{u}}_{k + 2}^{I} \in R_{k + 1, (β_{1})}^{I}$ for all $β_{1} \in (0, min {\sqrt{f^{″} (\pm 1)}})$ . In particular the $(k + 1)$ th inner order is computed and it holds $G_{k} \in R_{k, (β, γ)}^{C}$ as well as $g_{k + 1} \in {\hat{R}}_{k + 1, (β)}^{I} + R_{k, (β)}^{C}$ for all $β \in (0, min {\overline{β} (γ), \sqrt{f^{″} (\pm 1)}})$ , $γ \in (0, \overline{γ})$ , where $\overline{β}, \overline{γ}$ are as in Theorem 4.8. As in the last subparagraph we obtain a unique smooth solution ${\overline{u}}_{k + 1}^{C}$ to (5.24)–(5.25) with the decay ${\hat{u}}_{k + 1}^{C} \in R_{k, (β, γ)}^{C}$ for all $(β, γ)$ as above. Altogether, the $(k + 1)$ th order is constructed.

Finally, by induction the jth order is determined for all $j = 0, \dots, M$ , the $h_{j}$ are smooth and ${\hat{u}}_{j + 1}^{I} \in R_{j, (β_{1})}^{I}$ for all $β_{1} \in (0, min {\sqrt{f^{″} (\pm 1)}})$ as well as ${\hat{u}}_{j}^{C} \in R_{j, (β, γ)}^{C}$ for every $β \in (0, min {\overline{β} (γ), \sqrt{f^{″} (\pm 1)}})$ , $γ \in (0, \overline{γ})$ , where $\overline{β}, \overline{γ}$ are as in Theorem 4.8.
5.1.3. The approximate solution for (AC) in ND

Let $N ⩾ 2$ and $Γ : = {(Γ_{t})}_{t \in [0, T]}$ be as in Section 3.1 with contact angle $\frac{π}{2}$ and a solution to (MCF) in Ω. Moreover, let $δ > 0$ be such that the assertions of Theorem 3.2 hold for $2 δ$ instead of δ and let $r, s, b, σ, μ_{1}$ be as in the theorem. Furthermore, let $M \in N$ , $M ⩾ 2$ be the parameter used from the beginning of Section 5.1. Let $η : R \to [0, 1]$ be smooth with $η (r) = 1$ for $| r | ⩽ 1$ and $η (r) = 0$ for $| r | ⩾ 2$ . Then for $ε > 0$ we set $\begin{matrix} u_{ε}^{A} : = \{\begin{matrix} η (\frac{r}{δ}) [u_{ε}^{I} + u_{ε}^{C} η (\frac{b}{μ_{1}})] + (1 - η (\frac{r}{δ})) sign (r) & in \overline{Γ (2 δ)}, \\ \pm 1 & in Q_{T}^{\pm} ∖ Γ (2 δ), \end{matrix} \end{matrix}$ where $u_{ε}^{I}$ and $u_{ε}^{C}$ were constructed in Sections 5.1.1 and 5.1.2. This yields an approximate solution for (AC1)–(AC3) in the following sense:

Lemma 5.10.
The function $u_{ε}^{A}$ is smooth, uniformly bounded with respect to $x, t, ε$ and for the remainder $r_{ε}^{A} : = \partial_{t} u_{ε}^{A} - Δ u_{ε}^{A} + \frac{1}{ε^{2}} f^{'} (u_{ε}^{A})$ in ( AC1 ) and $s_{ε}^{A} : = \partial_{N_{\partial Ω}} u_{ε}^{A}$ in ( AC2 ) it holds $\begin{array}{l} | r_{ε}^{A} | ⩽ C (ε^{M} e^{- c | ρ_{ε} |} + ε^{M + 1}) in Γ (2 δ, μ_{1}), \\ | r_{ε}^{A} | ⩽ C (ε^{M - 1} e^{- c (| ρ_{ε} | + H_{ε})} + ε^{M} e^{- c | ρ_{ε} |} + ε^{M + 1}) in Γ^{C} (2 δ, 2 μ_{1}), \\ r_{ε}^{A} = 0 in Q_{T} ∖ Γ (2 δ), \\ | s_{ε}^{A} | ⩽ C ε^{M} e^{- c | ρ_{ε} |} on \partial Q_{T} \cap Γ (2 δ), \\ s_{ε}^{A} = 0 on \partial Q_{T} ∖ Γ (2 δ) \end{array}$ for $ε > 0$ small and some $c, C > 0$ . Here $ρ_{ε}$ is defined in ( 5.1 ) and $H_{ε} = \frac{b}{ε}$ .
Remark 5.11.
The estimate also holds without the $ε^{M + 1}$ -term. This follows from a more precise consideration of the remainder terms in the Taylor expansions in Sections 5.1.1–5.1.2 and below. Moreover, one could also lower the number of terms needed in the Taylor expansions a little bit by looking closely at the construction in the previous sections. This would only be of interest if one considers hypersurfaces of class $C^{l}$ for some large but finite l.
Proof of Lemma 5.10.
The third and the last equation are evident from the construction. Moreover, the rigorous Taylor expansions (5.2)–(5.3), (5.12)–(5.14) and (5.17) together with the remarks for the remainders and Sections 5.1.1–5.1.2 yield $\begin{array}{l} | \partial_{t} u_{ε}^{I} - Δ u_{ε}^{I} + \frac{1}{ε^{2}} f^{'} (u_{ε}^{I}) | ⩽ C (ε^{M} e^{- c | ρ_{ε} |} + ε^{M + 1}) in Γ (2 δ), \\ | \partial_{t} u_{ε}^{C} - Δ u_{ε}^{C} + \frac{f^{'} (u_{ε}^{I} + u_{ε}^{C}) - f^{'} (u_{ε}^{I})}{ε^{2}} | ⩽ C (ε^{M - 1} e^{- c (| ρ_{ε} | + H_{ε})} + ε^{M + 1}) in Γ^{C} (2 δ, 2 μ_{1}), \\ | \partial_{N_{\partial Ω}} (u_{ε}^{I} + u_{ε}^{C}) | ⩽ C ε^{M} e^{- c | ρ_{ε} |} on Γ^{C} (2 δ, 2 μ_{1}) \cap \partial Q_{T} . \end{array}$ Therefore the first estimate in the lemma holds for the remainder of $u_{ε}^{I}$ in (AC1) on $Γ (2 δ)$ and the second estimate in the lemma is true for the remainder of $u_{ε}^{I} + u_{ε}^{C}$ in (AC1) on $Γ^{C} (2 δ, 2 μ_{1})$ . In order to use this for $u_{ε}^{A}$ , we have to deal with the mixed terms due to the cutoff-functions. First, we prove that the remainder of ${\tilde{u}}_{ε}^{A} : = u_{ε}^{I} + u_{ε}^{C} η (b / μ_{1})$ in (AC1) satisfies the first two estimates in the lemma. Note that on $Γ (2 δ, 1 - 2 μ_{1})$ and $Γ^{C} (2 δ, μ_{1})$ this follows from the above estimates. Moreover, due to Taylor expansions it holds $\begin{array}{l} f^{'} (u_{ε}^{I} + η (\frac{b}{μ_{1}}) u_{ε}^{C}) & = f^{'} (u_{ε}^{I}) + O (η (\frac{b}{μ_{1}}) | u_{ε}^{C} |) \\ = (1 - η (\frac{b}{μ_{1}})) f^{'} (u_{ε}^{I}) + η (\frac{b}{μ_{1}}) f^{'} (u_{ε}^{I} + u_{ε}^{C}) + O (η (\frac{b}{μ_{1}}) | u_{ε}^{C} |) . \end{array}$ Hence with the product rule for $\partial_{t}$ and Δ we obtain $\begin{array}{l} [l.h.s. in (AC1) for u_{ε}^{I} + η (\frac{b}{μ_{1}}) u_{ε}^{C}] \\ = (1 - η (\frac{b}{μ_{1}})) [l.h.s. in (AC1) for u_{ε}^{I}] + η (\frac{b}{μ_{1}}) [l.h.s. in (AC1) for u_{ε}^{I} + u_{ε}^{C}] \\ + u_{ε}^{C} (\partial_{t} - Δ) [η (\frac{b}{μ_{1}})] - 2 \nabla [η (\frac{b}{μ_{1}})] \cdot \nabla u_{ε}^{C} + \frac{1}{ε^{2}} O (η (\frac{b}{μ_{1}}) | u_{ε}^{C} |), \end{array}$ where “l.h.s.” stands for “left hand side”. Hence the above estimates and the asymptotics of ${\hat{u}}_{j}^{C}$ for $j = 0, \dots, M$ yield the desired estimates for the remainder of ${\tilde{u}}_{ε}^{A}$ in (AC1) on $Γ (2 δ)$ . Altogether the first two estimates hold for δ instead of $2 δ$ . In $Γ (2 δ) ∖ Γ (δ)$ we have again similar mixed terms as above due to the cutoff functions. With Taylor expansions we obtain in $Q_{T}^{\pm} ∖ Γ (δ)$ : $\begin{matrix} f^{'} (u_{ε}^{A}) = O (η (\frac{r}{δ}) | {\tilde{u}}_{ε}^{A} \mp 1 |), η (\frac{r}{δ}) f^{'} ({\tilde{u}}_{ε}^{A}) = O (η (\frac{r}{δ}) | {\tilde{u}}_{ε}^{A} \mp 1 |), \end{matrix}$ where we used $f^{'} (\pm 1) = 0$ . Hence the product rule for $\partial_{t}$ and Δ yields in $Γ (2 δ) ∖ Γ (δ)$ $\begin{array}{l} [l.h.s. in (AC1) for u_{ε}^{A}] & = η (\frac{r}{δ}) [l.h.s. in (AC1) for {\tilde{u}}_{ε}^{A}] + ({\tilde{u}}_{ε}^{A} \mp 1) (\partial_{t} - Δ) [η (\frac{r}{δ})] \\ - 2 \nabla [η (\frac{r}{δ})] \cdot \nabla {\tilde{u}}_{ε}^{A} + \frac{1}{ε^{2}} O (η (\frac{r}{δ}) | {\tilde{u}}_{ε}^{A} \mp 1 |) . \end{array}$ Finally, the asymptotics of $u_{j}^{I}$ and $u_{j}^{C}$ for $j = 0, \dots, M$ imply the estimates for $r_{ε}^{A}$ .

It is left to prove the remaining assertion for $s_{ε}^{A}$ . By definition it holds $u_{ε}^{A} = u_{ε}^{I} + u_{ε}^{C}$ on $Γ^{C} (δ, μ_{1}) \cap \partial Q_{T}$ . For the latter we already have an estimate of the Neumann derivative on $Γ^{C} (2 δ, μ_{1}) \cap \partial Q_{T}$ , see above. Again we have mixed terms for $u_{ε}^{A}$ on $[Γ^{C} (2 δ, μ_{1}) \cap \partial Q_{T}] ∖ Γ (δ)$ because of the cutoff-functions. On the latter set it holds ${\tilde{u}}_{ε}^{A} = u_{ε}^{I} + u_{ε}^{C}$ and $\begin{matrix} \partial_{N_{\partial Ω}} u_{ε}^{A} = ({\tilde{u}}_{ε}^{A} \mp 1) \partial_{N_{\partial Ω}} [η (\frac{r}{ε})] + η (\frac{r}{ε}) \partial_{N_{\partial Ω}} {\tilde{u}}_{ε}^{A} . \end{matrix}$ Therefore the claim follows with the asymptotics of $u_{j}^{I}$ and $u_{j}^{C}$ for $j = 0, \dots, M$ . □

5.2. Asymptotic expansion of (vAC) in ND

Let $N ⩾ 2$ , $Ω \subseteq R^{N}$ , $Γ : = {(Γ_{t})}_{t \in [0, T]}$ and $δ > 0$ be as in the beginning of Section 5.1, in particular Γ is a smooth solution to (MCF) with 90°-contact angle condition in Ω. Moreover, let $W : R^{m} \to R$ be as in Definition 1.1 and ${\vec{u}}_{\pm}$ be any distinct pair of minimizers of W. In this section we construct a smooth approximate solution ${\vec{u}}_{ε}^{A}$ to (vAC1)–(vAC3) with ${\vec{u}}_{ε}^{A} = {\vec{u}}_{\pm}$ in $Q_{T}^{\pm} ∖ Γ (2 δ)$ , increasingly “steep” transition from ${\vec{u}}_{-}$ to ${\vec{u}}_{+}$ for $ε \to 0$ and such that the set ${{\vec{u}}_{ε}^{A} \in \frac{1}{2} ({\vec{u}}_{+} + {\vec{u}}_{-}) + span {{\vec{u}}_{+} - {\vec{u}}_{-}}}^{⊥}$ converges to Γ for $ε \to 0$ . All computations are very similar to the ones in Section 5.1. We just have to incorporate vector-valued functions and for the appearing vector-valued model problems we use the corresponding solution theorems in Sections 4.3–4.4. For the latter we make the assumption $dim ker {\overset{ˇ}{L}}_{0} = 1$ , where ${\overset{ˇ}{L}}_{0}$ is as in Remark 4.11 for a solution ${\vec{θ}}_{0}$ as in Theorem 4.9.

Let $M \in N$ with $M ⩾ 2$ . Again we introduce height functions $\begin{matrix} {\overset{ˇ}{h}}_{j} : Σ \times [0, T] \to R for j = 1, \dots, M and {\overset{ˇ}{h}}_{ε} : = \sum_{j = 1}^{M} ε^{j - 1} {\overset{ˇ}{h}}_{j} . \end{matrix}$ Moreover, we set ${\overset{ˇ}{h}}_{M + 1} : = {\overset{ˇ}{h}}_{M + 2} : = 0$ and we define the scaled variable $\begin{matrix} (5.27) & {\overset{ˇ}{ρ}}_{ε} (x, t) : = \frac{r (x, t)}{ε} - {\overset{ˇ}{h}}_{ε} (s (x, t), t) for (x, t) \in \overline{Γ (2 δ)} . \end{matrix}$

In Section 5.2.1 we construct the inner expansion and in Section 5.2.2 the contact point expansion. Finally, in Section 5.2.3 the result on the approximation error of ${\vec{u}}_{ε}^{A}$ can be found.

5.2.1. Inner expansion of (vAC) in ND

For the inner expansion we consider the following ansatz: Let $ε > 0$ be small and $\begin{matrix} {\vec{u}}_{ε}^{I} : = \sum_{j = 0}^{M + 1} ε^{j} {\vec{u}}_{j}^{I}, {\vec{u}}_{j}^{I} (x, t) : = {\overset{ˇ}{u}}_{j}^{I} (ρ_{ε} (x, t), s (x, t), t) for (x, t) \in \overline{Γ (2 δ)}, \end{matrix}$ where $\begin{matrix} {\overset{ˇ}{u}}_{j}^{I} : R \times Σ \times [0, T] \to R^{m} : (ρ, s, t) \mapsto {\overset{ˇ}{u}}_{j}^{I} (ρ, s, t) \end{matrix}$ for $j = 0, \dots, M + 1$ . Moreover, we set ${\vec{u}}_{M + 2}^{I} : = 0$ and ${\overset{ˇ}{u}}_{ε}^{I} : = \sum_{j = 0}^{M + 1} ε^{j} {\overset{ˇ}{u}}_{j}^{I}$ . We will expand (vAC1) for ${\vec{u}}_{ε} = {\vec{u}}_{ε}^{I}$ into ε-series with coefficients in $({\overset{ˇ}{ρ}}_{ε}, s, t)$ up to $O (ε^{M - 1})$ . This yields equations of analogous form as in the scalar case in Section 5.1.1. Therefore we have to compute the action of the differential operators on ${\vec{u}}_{ε}^{I}$ .

In the following we use the same conventions as in Lemma 5.1. Moreover, for a sufficiently smooth $\vec{g} : Σ \to R^{m}$ we set $D_{Σ} \vec{g} : = {(\nabla_{Σ} g_{1}, \dots, \nabla_{Σ} g_{m})}^{⊤} : Σ \to R^{m \times N}$ .

Lemma 5.12.
Let $ε > 0$ , $\overset{ˇ}{w} : R \times Σ \times [0, T] \to R^{m}$ be sufficiently smooth and $\vec{w} : \overline{Γ (2 δ)} \to R^{m}$ be defined by $\vec{w} (x, t) : = \overset{ˇ}{w} ({\overset{ˇ}{ρ}}_{ε} (x, t), s (x, t), t)$ for all $(x, t) \in \overline{Γ (2 δ)}$ . Then it holds $\begin{array}{l} \partial_{t} \vec{w} = \partial_{ρ} \overset{ˇ}{w} [\frac{\partial_{t} r}{ε} - (\partial_{t} {\overset{ˇ}{h}}_{ε} + \partial_{t} s \cdot \nabla_{Σ} {\overset{ˇ}{h}}_{ε})] + D_{Σ} \overset{ˇ}{w} \partial_{t} s + \partial_{t} \overset{ˇ}{w}, \\ D_{x} \vec{w} = \partial_{ρ} \overset{ˇ}{w} {[\frac{\nabla r}{ε} - {(D_{x} s)}^{⊤} \nabla_{Σ} {\overset{ˇ}{h}}_{ε}]}^{⊤} + D_{Σ} \overset{ˇ}{w} D_{x} s, \\ \begin{matrix} Δ \vec{w} & = \partial_{ρ} \overset{ˇ}{w} [\frac{Δ r}{ε} - (Δ s \cdot \nabla_{Σ} {\overset{ˇ}{h}}_{ε} + \sum_{i, l = 1}^{N} \nabla s_{i} \cdot \nabla s_{l} {(\nabla_{Σ})}_{i} {(\nabla_{Σ})}_{l} {\overset{ˇ}{h}}_{ε})] \\ + D_{Σ} \overset{ˇ}{w} Δ s + \sum_{i, l = 1}^{N} \nabla s_{i} \cdot \nabla s_{l} {(\nabla_{Σ})}_{i} {(\nabla_{Σ})}_{l} \overset{ˇ}{w} \\ + 2 D_{Σ} \partial_{ρ} \overset{ˇ}{w} D_{x} s [\frac{\nabla r}{ε} - {(D_{x} s)}^{⊤} \nabla_{Σ} {\overset{ˇ}{h}}_{ε}] + \partial_{ρ}^{2} \overset{ˇ}{w} {| \frac{\nabla r}{ε} - {(D_{x} s)}^{⊤} \nabla_{Σ} {\overset{ˇ}{h}}_{ε} |}^{2}, \end{matrix} \end{array}$ where the $\vec{w}$ -terms on the left hand side and derivatives of r or s are evaluated at $(x, t) \in \overline{Γ (2 δ)}$ , the ${\overset{ˇ}{h}}_{ε}$ -terms at $(s (x, t), t)$ and the $\overset{ˇ}{w}$ -terms at $({\overset{ˇ}{ρ}}_{ε} (x, t), s (x, t), t)$ .
Proof of Lemma 5.12.
This follows from Lemma 5.1 applied to every component. □

For the expansion of (vAC1) for ${\vec{u}}_{ε} = {\vec{u}}_{ε}^{I}$ we use Taylor expansions again. For the $\nabla W$ -part this yields: If the ${\vec{u}}_{j}^{I}$ are bounded, then $\begin{matrix} (5.28) & \nabla W ({\vec{u}}_{ε}^{I}) = \nabla W ({\vec{u}}_{0}^{I}) + \sum_{ν \in N_{0}^{m}, | ν | = 1}^{M + 2} \frac{\partial_{y}^{ν} \nabla W ({\vec{u}}_{0}^{I})}{ν!} {[\sum_{j = 1}^{M + 1} {\vec{u}}_{j}^{I} ε^{j}]}^{ν} + O (ε^{M + 3}) on \overline{Γ (2 δ)} . \end{matrix}$ The terms in the ε-expansion that are needed explicitly are $\begin{array}{l} O (1) : \nabla W ({\vec{u}}_{0}^{I}), \\ O (ε) : D^{2} W ({\vec{u}}_{0}^{I}) {\vec{u}}_{1}^{I}, \\ O (ε^{2}) : D^{2} W ({\vec{u}}_{0}^{I}) {\vec{u}}_{2}^{I} + \sum_{ν \in N_{0}^{m}, | ν | = 2} \frac{\partial_{y}^{ν} \nabla W ({\vec{u}}_{0}^{I})}{ν!} {[{\vec{u}}_{1}^{I}]}^{ν} . \end{array}$ For $k = 3, \dots, M + 2$ the order $O (ε^{k})$ is given by $\begin{array}{l} O (ε^{k}) : D^{2} W ({\vec{u}}_{0}^{I}) {\vec{u}}_{k}^{I} + [ & some polynomial in entries of ({\vec{u}}_{1}^{I}, \dots, {\vec{u}}_{k - 1}^{I}) of order ⩽ k, \\ where the coefficients are multiples of \partial_{y}^{ν} \nabla W ({\vec{u}}_{0}^{I}), \\ ν \in N_{0}^{m}, | ν | = 2, \dots, k and every term contains a {({\vec{u}}_{j}^{I})}_{n} -factor] . \end{array}$ The other explicit terms in (5.28) are of order $O (ε^{M + 3})$ .

Moreover, we expand functions of $(x, t) \in \overline{Γ (2 δ)}$ into ε-series analogously to the scalar case, cf. the Taylor expansion (5.3) and the remarks there. We just replace $h_{j}, ρ_{ε}$ by ${\overset{ˇ}{h}}_{j}, {\overset{ˇ}{ρ}}_{ε}$ .

For the higher orders in the expansion we use analogous definitions as in the scalar case:
Definition 5.13 (Notation for Inner Expansion of (vAC)).

We call $({\vec{θ}}_{0}, {\vec{u}}_{1}^{I})$ the zero-th inner order and $({\overset{ˇ}{h}}_{j}, {\vec{u}}_{j + 1}^{I})$ the jth inner order for $j = 1, \dots, M$ .

Let $k \in {- 1, \dots, M + 2}$ and $β > 0$ . We denote with ${\overset{ˇ}{R}}_{k, (β)}^{I}$ the set of smooth vector-valued functions $\vec{R} : R \times Σ \times [0, T] \to R^{m}$ that depend only on the jth inner orders for $0 ⩽ j ⩽ min {k, M}$ and satisfy uniformly in $(ρ, s, t)$ : $\begin{matrix} | \partial_{ρ}^{i} {(\nabla_{Σ})}_{n_{1}} \dots {(\nabla_{Σ})}_{n_{d}} \partial_{t}^{n} \vec{R} (ρ, s, t) | = O (e^{- β | ρ |}) \end{matrix}$ for all $n_{1}, \dots, n_{d} \in {1, \dots, N}$ and $d, i, n \in N_{0}$ .

For $k \in {- 1, \dots, M + 2}$ and $β > 0$ the set ${\tilde{R}}_{k, (β)}^{I}$ is defined analogously to ${\overset{ˇ}{R}}_{k, (β)}^{I}$ with functions of type $\vec{R} : R \times \partial Σ \times [0, T] \to R^{m}$ instead.

Now we expand (vAC1) for ${\vec{u}}_{ε} = {\vec{u}}_{ε}^{I}$ into ε-series. This is analogous to the scalar case in Section 5.1.1. In the following $(ρ, s, t)$ are always in $R \times Σ \times [0, T]$ and sometimes omitted.

Inner expansion: $O (ε^{- 2})$ . Using $| \nabla r |^{2} |_{{\overline{X}}_{0} (s, t)} = 1$ due to Theorem 3.2, the $O (\frac{1}{ε^{2}})$ -order is zero if $\begin{matrix} (5.29) & - \partial_{ρ}^{2} {\overset{ˇ}{u}}_{0}^{I} (ρ, s, t) + \nabla W ({\overset{ˇ}{u}}_{0}^{I} (ρ, s, t)) = 0 . \end{matrix}$ Since we want to connect the minima ${\vec{u}}_{\pm}$ of W, we require ${lim}_{ρ \to \pm \infty} {\overset{ˇ}{u}}_{0}^{I} (ρ, s, t) = {\vec{u}}_{\pm}$ . Moreover, it is natural to ask for $R_{{\vec{u}}_{-}, {\vec{u}}_{+}} {\overset{ˇ}{u}}_{0}^{I} |_{ρ = 0} = {\overset{ˇ}{u}}_{0}^{I} |_{ρ = 0}$ , since then ${\overset{ˇ}{u}}_{0}^{I} |_{ρ = 0}$ can be interpreted as being in the middle of the two phases ${\vec{u}}_{\pm}$ . Here $R_{{\vec{u}}_{-}, {\vec{u}}_{+}}$ is as in Definition 1.1. By Theorem 4.9 there is a smooth $R_{{\vec{u}}_{-}, {\vec{u}}_{+}}$ -odd ${\vec{θ}}_{0} : R \to R^{m}$ such that ${\overset{ˇ}{u}}_{0}^{I} (ρ, s, t) : = {\vec{θ}}_{0} (ρ)$ solves (5.29) and $\begin{matrix} \partial_{z}^{l} [{\vec{θ}}_{0} - {\vec{u}}_{\pm}] (ρ) = O (e^{- β | ρ |}) for ρ \to \pm \infty and all l \in N_{0}, β \in (0, \sqrt{λ / 2}), \end{matrix}$ where $λ > 0$ is such that $D^{2} W ({\vec{u}}_{\pm}) ⩾ λ I$ . Moreover, it holds $R_{{\vec{u}}_{-}, {\vec{u}}_{+}} {\vec{θ}}_{0}^{'} |_{ρ = 0} \neq {\vec{θ}}_{0}^{'} |_{ρ = 0}$ .

Inner expansion: $O (ε^{- 1})$ . Analogously to the scalar case, cf. Section 5.1.1, it follows that the $O (\frac{1}{ε})$ -order cancels if $\begin{matrix} {\overset{ˇ}{L}}_{0} {\overset{ˇ}{u}}_{1}^{I} (ρ, s, t) + {\vec{θ}}_{0}^{'} (ρ) (\partial_{t} r - Δ r) |_{{\overline{X}}_{0} (s, t)} = 0, where {\overset{ˇ}{L}}_{0} : = - \partial_{ρ}^{2} + D^{2} W ({\vec{θ}}_{0}) . \end{matrix}$ Moreover, for convenience we require that ${({\overset{ˇ}{u}}_{1}^{I} |_{ρ = 0}, {\vec{θ}}_{0}^{'})}_{L^{2} {(R)}^{m}} = 0$ . We assume $dim ker {\overset{ˇ}{L}}_{0} = 1$ with respect to the spaces in (4.13), cf. Remark 4.11. Then due to Theorem 4.13 and Remark 4.14, 1. this parameter-dependent ODE together with the additional condition and suitable decay in $| ρ |$ has a unique solution ${\overset{ˇ}{u}}_{1}^{I}$ if and only if $(\partial_{t} r - Δ r) |_{{\overline{X}}_{0} (s, t)} = 0$ . The latter holds since it is equivalent to (MCF) for Γ by Theorem 3.2. Therefore we set ${\overset{ˇ}{u}}_{1}^{I} : = 0$ .

Inner expansion: $O (ε^{0})$ . In the analogous way as in the scalar case, cf. Section 5.1.1, the $O (1)$ -term in the expansion cancels if we require $\begin{array}{l} (5.30) & - {\overset{ˇ}{L}}_{0} {\overset{ˇ}{u}}_{2}^{I} (ρ, s, t) = {\vec{R}}_{1} (ρ, s, t), \\ \begin{matrix} {\vec{R}}_{1} (ρ, s, t) : = & {\vec{θ}}_{0}^{'} (ρ) [- \partial_{t} {\overset{ˇ}{h}}_{1} + \sum_{i, l = 1}^{N} \nabla s_{i} \cdot \nabla s_{l} {(\nabla_{Σ})}_{i} {(\nabla_{Σ})}_{l} {\overset{ˇ}{h}}_{1} \\ + (ρ + {\overset{ˇ}{h}}_{1}) \partial_{r} ((\partial_{t} r - Δ r) \circ \overline{X}) |_{(0, s, t)} - (\partial_{t} s - Δ s) |_{{\overline{X}}_{0} (s, t)} \cdot \nabla_{Σ} {\overset{ˇ}{h}}_{1}] \\ + {\vec{θ}}_{0}^{″} (ρ) [- \frac{1}{2} {(ρ + {\overset{ˇ}{h}}_{1})}^{2} \partial_{r}^{2} (| \nabla r |^{2} \circ \overline{X}) |_{(0, s, t)} \\ + 2 (ρ + {\overset{ˇ}{h}}_{1}) \partial_{r} ({(D_{x} s \nabla r)}^{⊤} \circ \overline{X}) |_{(0, s, t)} \nabla_{Σ} {\overset{ˇ}{h}}_{1} - {| {(D_{x} s)}^{⊤} |_{{\overline{X}}_{0} (s, t)} \nabla_{Σ} {\overset{ˇ}{h}}_{1} |}^{2}] . \end{matrix} \end{array}$ If ${\overset{ˇ}{h}}_{1}$ is smooth, then ${\vec{R}}_{1}$ is smooth and together with all derivatives decays exponentially in $| ρ |$ uniformly in $(s, t)$ with rate β for every $β \in (0, \sqrt{λ / 2})$ because of Theorem 4.9. Therefore Theorem 4.13 (applied in local coordinates for Σ) yields that there is a unique solution ${\overset{ˇ}{u}}_{2}^{I}$ to (5.30) together with suitable regularity and decay as well as ${({\overset{ˇ}{u}}_{2}^{I} |_{ρ = 0}, {\vec{θ}}_{0}^{'})}_{L^{2} {(R)}^{m}} = 0$ if and only if $\int_{R} {\vec{R}}_{1} (ρ, s, t) \cdot {\vec{θ}}_{0}^{'} (ρ) d ρ = 0$ . Because of integration by parts it holds $\int_{R} {\vec{θ}}_{0}^{'} (ρ) \cdot {\vec{θ}}_{0}^{″} (ρ) d ρ = 0$ . Therefore the nonlinearities in ${\overset{ˇ}{h}}_{1}$ cancel and we obtain a linear non-autonomous parabolic equation for ${\overset{ˇ}{h}}_{1}$ on Σ: $\begin{matrix} (5.31) & \partial_{t} {\overset{ˇ}{h}}_{1} - \sum_{i, l = 1}^{N} \nabla s_{i} \cdot \nabla s_{l} |_{{\overline{X}}_{0} (s, t)} {(\nabla_{Σ})}_{i} {(\nabla_{Σ})}_{l} {\overset{ˇ}{h}}_{1} + {\overset{ˇ}{a}}_{1} \cdot \nabla_{Σ} {\overset{ˇ}{h}}_{1} + {\overset{ˇ}{a}}_{0} {\overset{ˇ}{h}}_{1} = {\overset{ˇ}{f}}_{0} \end{matrix}$ in $Σ \times [0, T]$ . Here with $\begin{array}{l} {\overset{ˇ}{d}}_{1} : = \int_{R} {| {\vec{θ}}_{0}^{'} (ρ) |}^{2} d ρ, {\overset{ˇ}{d}}_{2} : = \int_{R} {| {\vec{θ}}_{0}^{'} (ρ) |}^{2} ρ d ρ, {\overset{ˇ}{d}}_{3} : = \int_{R} {| {\vec{θ}}_{0}^{'} (ρ) |}^{2} ρ^{2} d ρ, \\ {\overset{ˇ}{d}}_{4} : = \int_{R} {\vec{θ}}_{0}^{'} (ρ) \cdot {\vec{θ}}_{0}^{″} (ρ) ρ d ρ, {\overset{ˇ}{d}}_{5} : = \int_{R} {\vec{θ}}_{0}^{'} (ρ) \cdot {\vec{θ}}_{0}^{″} (ρ) ρ^{2} d ρ, {\overset{ˇ}{d}}_{6} : = \int_{R} {\vec{θ}}_{0}^{'} (ρ) \cdot {\vec{θ}}_{0}^{″} (ρ) ρ^{3} d ρ, \end{array}$ we have defined for all $(s, t) \in Σ \times [0, T]$ : $\begin{array}{l} (5.32) & {\overset{ˇ}{a}}_{1} (s, t) : = (\partial_{t} s - Δ s) |_{{\overline{X}}_{0} (s, t)} - 2 \frac{{\overset{ˇ}{d}}_{4}}{{\overset{ˇ}{d}}_{1}} \partial_{r} ({(D_{x} s \nabla r)}^{⊤} \circ \overline{X}) |_{(0, s, t)} \in R^{N}, \\ (5.33) & {\overset{ˇ}{a}}_{0} (s, t) : = - \partial_{r} ((\partial_{t} r - Δ r) \circ \overline{X}) |_{(0, s, t)} + \frac{{\overset{ˇ}{d}}_{4}}{{\overset{ˇ}{d}}_{1}} \partial_{r}^{2} (| \nabla r |^{2} \circ \overline{X}) |_{(0, s, t)} \in R, \\ (5.34) & {\overset{ˇ}{f}}_{0} (s, t) : = \frac{{\overset{ˇ}{d}}_{2}}{{\overset{ˇ}{d}}_{1}} \partial_{r} ((\partial_{t} r - Δ r) \circ \overline{X}) |_{(0, s, t)} - \frac{{\overset{ˇ}{d}}_{5}}{2 {\overset{ˇ}{d}}_{1}} \partial_{r}^{2} (| \nabla r |^{2} \circ \overline{X}) |_{(0, s, t)} \in R . \end{array}$ Note that since ${\vec{θ}}_{0}$ is $R_{{\vec{u}}_{-}, {\vec{u}}_{+}}$ -odd and due to the isometry properties of $R_{{\vec{u}}_{-}, {\vec{u}}_{+}}$ , it follows that ${\overset{ˇ}{d}}_{2} = {\overset{ˇ}{d}}_{5} = 0$ and hence also ${\overset{ˇ}{f}}_{0} = 0$ . Therefore the equation (5.31) for ${\overset{ˇ}{h}}_{1}$ is homogeneous. Note that this corresponds to the case of symmetric f in the scalar case, cf. Remark 5.5. This is due to the fact that we restricted to symmetric W in the vector-valued case.

If ${\overset{ˇ}{h}}_{1}$ is smooth and solves (5.31), then Theorem 4.13 (applied in local coordinates for Σ) yields a smooth solution ${\overset{ˇ}{u}}_{2}^{I}$ to (5.30) with ${({\overset{ˇ}{u}}_{2}^{I} |_{ρ = 0}, {\vec{θ}}_{0}^{'})}_{L^{2} {(R)}^{m}} = 0$ and we get decay estimates. With Remark 5.4 and compactness we obtain ${\overset{ˇ}{u}}_{2}^{I} \in {\overset{ˇ}{R}}_{1, (β)}^{I}$ for any $β \in (0, min {\sqrt{λ / 2}, {\overset{ˇ}{β}}_{0}})$ , where ${\overset{ˇ}{β}}_{0} > 0$ is as in Theorem 4.13.

Inner expansion: $O (ε^{k})$ . Let $k \in {1, \dots, M - 1}$ and suppose that the jth inner order has already been constructed for $j = 0, \dots, k$ , that it is smooth and ${\overset{ˇ}{u}}_{j + 1}^{I} \in {\overset{ˇ}{R}}_{j, (β)}^{I}$ for every $β \in (0, min {\sqrt{λ / 2}, {\overset{ˇ}{β}}_{0}})$ with ${\overset{ˇ}{β}}_{0} > 0$ as in Theorem 4.13. Analogously to the scalar case one can compute the $O (ε^{k})$ -order in (vAC1) for ${\vec{u}}_{ε} = {\vec{u}}_{ε}^{I}$ . This yields that the order $O (ε^{k})$ is zero if $\begin{array}{l} (5.35) & - {\overset{ˇ}{L}}_{0} {\overset{ˇ}{u}}_{k + 2}^{I} (ρ, s, t) = {\vec{R}}_{k + 1} (ρ, s, t), \\ \begin{matrix} {\vec{R}}_{k + 1} (ρ, s, t) : = & {\vec{θ}}_{0}^{'} (ρ) [- \partial_{t} {\overset{ˇ}{h}}_{k + 1} + \sum_{i, l = 1}^{N} \nabla s_{i} \cdot \nabla s_{l} |_{{\overline{X}}_{0} (s, t)} {(\nabla_{Σ})}_{i} {(\nabla_{Σ})}_{l} {\overset{ˇ}{h}}_{k + 1} \\ - (\partial_{t} s - Δ s) |_{{\overline{X}}_{0} (s, t)} \cdot \nabla_{Σ} {\overset{ˇ}{h}}_{k + 1} + {\overset{ˇ}{h}}_{k + 1} \partial_{r} ((\partial_{t} r - Δ r) \circ \overline{X}) |_{(0, s, t)}] \\ + {\vec{θ}}_{0}^{″} (ρ) [- (ρ + {\overset{ˇ}{h}}_{1}) {\overset{ˇ}{h}}_{k + 1} \partial_{r}^{2} (| \nabla r |^{2} \circ \overline{X}) |_{(0, s, t)} \\ - 2 {(\nabla_{Σ} {\overset{ˇ}{h}}_{1})}^{⊤} D_{x} s {(D_{x} s)}^{⊤} |_{{\overline{X}}_{0} (s, t)} \nabla_{Σ} {\overset{ˇ}{h}}_{k + 1} \\ + 2 \partial_{r} ({(D_{x} s \nabla r)}^{⊤} \circ \overline{X}) |_{(0, s, t)} [(ρ + {\overset{ˇ}{h}}_{1}) \nabla_{Σ} {\overset{ˇ}{h}}_{k + 1} + {\overset{ˇ}{h}}_{k + 1} \nabla_{Σ} {\overset{ˇ}{h}}_{1}]] \\ + {\overset{ˇ}{R}}_{k} (ρ, s, t), \end{matrix} \end{array}$ where ${\overset{ˇ}{R}}_{k} \in {\overset{ˇ}{R}}_{k, (β)}^{I}$ . If ${\overset{ˇ}{h}}_{k + 1}$ is smooth, then due to Theorem 4.13 equation (5.35) admits a unique solution ${\overset{ˇ}{u}}_{k + 2}^{I}$ with suitable regularity and decay as well as ${({\overset{ˇ}{u}}_{2}^{I} |_{ρ = 0}, {\vec{θ}}_{0}^{'})}_{L^{2} {(R)}^{m}} = 0$ if and only if $\int_{R} {\vec{R}}_{k + 1} (ρ, s, t) \cdot {\vec{θ}}_{0}^{'} (ρ) d ρ = 0$ . Because of $\int_{R} {\vec{θ}}_{0}^{″} \cdot {\vec{θ}}_{0}^{'} = 0$ , the latter is equivalent to $\begin{matrix} (5.36) & \partial_{t} {\overset{ˇ}{h}}_{k + 1} - \sum_{i, l = 1}^{N} \nabla s_{i} \cdot \nabla s_{l} |_{{\overline{X}}_{0} (s, t)} {(\nabla_{Σ})}_{i} {(\nabla_{Σ})}_{l} {\overset{ˇ}{h}}_{k + 1} + {\overset{ˇ}{a}}_{1} \cdot \nabla_{Σ} {\overset{ˇ}{h}}_{k + 1} + {\overset{ˇ}{a}}_{0} {\overset{ˇ}{h}}_{k + 1} = {\overset{ˇ}{f}}_{k}, \end{matrix}$ where $\begin{matrix} {\overset{ˇ}{f}}_{k} (s, t) : = \int_{R} {\overset{ˇ}{R}}_{k} (ρ, s, t) \cdot {\vec{θ}}_{0}^{'} (ρ) d ρ / {‖ {\vec{θ}}_{0}^{'} ‖}_{L^{2} {(R)}^{m}}^{2} \end{matrix}$ is a smooth function of $(s, t)$ and depends only on the jth inner orders for $0 ⩽ j ⩽ k$ . Here ${\overset{ˇ}{a}}_{0}, {\overset{ˇ}{a}}_{1}$ are defined in (5.32)–(5.33). If ${\overset{ˇ}{h}}_{k + 1}$ is smooth and solves (5.36), then Theorem 4.13 yields as in Section 5.2.1 a smooth solution ${\overset{ˇ}{u}}_{k + 2}^{I}$ to (5.35) such that ${\overset{ˇ}{u}}_{k + 2}^{I} \in {\overset{ˇ}{R}}_{k + 1, (β)}^{I}$ for all $β \in (0, min {\sqrt{λ / 2}, {\overset{ˇ}{β}}_{0}})$ .

5.2.2. Contact point expansion of (vAC) in ND

This is analogous to the scalar case, cf. Section 5.1.2. We make the ansatz ${\vec{u}}_{ε} = {\vec{u}}_{ε}^{I} + {\vec{u}}_{ε}^{C}$ in $Γ (2 δ)$ close to the contact points. Let $σ, b : \overline{Γ^{C} (2 δ, 2 μ_{1})} \to \partial Σ \times [0, 2 μ_{1}]$ be as in Theorem 3.2. Then with $H_{ε} : = \frac{b}{ε}$ we set $\begin{matrix} {\vec{u}}_{ε}^{C} : = \sum_{j = 1}^{M} ε^{j} {\vec{u}}_{j}^{C}, {\vec{u}}_{j}^{C} (x, t) : = {\overset{ˇ}{u}}_{j}^{C} ({\overset{ˇ}{ρ}}_{ε} (x, t), H_{ε} (x, t), σ (x, t), t) for (x, t) \in \overline{Γ^{C} (2 δ, 2 μ_{1})}, \end{matrix}$ where $\begin{matrix} {\overset{ˇ}{u}}_{j}^{C} : \overline{R_{+}^{2}} \times \partial Σ \times [0, T] \to R^{m} : (ρ, H, σ, t) \mapsto {\overset{ˇ}{u}}_{j}^{C} (ρ, H, σ, t) \end{matrix}$ for $j = 1, \dots, M$ . Moreover, we define ${\vec{u}}_{M + 1}^{C} : = {\vec{u}}_{M + 2}^{C} : = 0$ and ${\overset{ˇ}{u}}_{ε}^{C} : = \sum_{j = 1}^{M} ε^{j} {\overset{ˇ}{u}}_{j}^{C}$ . As in the scalar case, instead of (vAC1) for ${\vec{u}}_{ε} = {\vec{u}}_{ε}^{I} + {\vec{u}}_{ε}^{C}$ , we will expand the “bulk equation” $\begin{matrix} (5.37) & \partial_{t} {\vec{u}}_{ε}^{C} - Δ {\vec{u}}_{ε}^{C} + \frac{1}{ε^{2}} [\nabla W ({\vec{u}}_{ε}^{I} + {\vec{u}}_{ε}^{C}) - \nabla W ({\vec{u}}_{ε}^{I})] = 0 \end{matrix}$ into ε-series with coefficients in $({\overset{ˇ}{ρ}}_{ε}, H_{ε}, σ, t)$ up to $O (ε^{M - 2})$ . Moreover, we will expand (vAC2) for ${\vec{u}}_{ε} = {\vec{u}}_{ε}^{I} + {\vec{u}}_{ε}^{C}$ into ε-series with coefficients in $({\overset{ˇ}{ρ}}_{ε}, σ, t)$ up to $O (ε^{M - 1})$ . Altogether we end up with analogous equations as in the scalar case. The solvability condition (4.18) will yield the boundary conditions on $\partial Σ \times [0, T]$ for the height functions ${\overset{ˇ}{h}}_{j}$ .

For the expansions we calculate the action of the differential operators on ${\vec{u}}_{ε}^{C}$ in the next lemma. Here we use the same conventions as in Lemma 5.12 and define $D_{\partial Σ}$ in an analogous way as $D_{Σ}$ .

Lemma 5.14.
Let $\overline{R_{+}^{2}} \times \partial Σ \times [0, T] ∋ (ρ, H, σ, t) \mapsto \overset{ˇ}{w} (ρ, H, σ, t) \in R^{m}$ be sufficiently smooth and let $\vec{w} : \overline{Γ^{C} (2 δ, 2 μ_{1})} \to R^{m} : (x, t) \mapsto \overset{ˇ}{w} ({\overset{ˇ}{ρ}}_{ε} (x, t), H_{ε} (x, t), σ (x, t), t)$ . Then $\begin{array}{l} \partial_{t} \vec{w} = \partial_{ρ} \overset{ˇ}{w} [\frac{\partial_{t} r}{ε} - (\partial_{t} {\overset{ˇ}{h}}_{ε} + \partial_{t} s \cdot \nabla_{Σ} {\overset{ˇ}{h}}_{ε})] + \partial_{H} \overset{ˇ}{w} \frac{\partial_{t} b}{ε} + D_{\partial Σ} \overset{ˇ}{w} \partial_{t} σ + \partial_{t} \overset{ˇ}{w}, \\ D_{x} \vec{w} = \partial_{ρ} \overset{ˇ}{w} {[\frac{\nabla r}{ε} - {(D_{x} s)}^{⊤} \nabla_{Σ} {\overset{ˇ}{h}}_{ε}]}^{⊤} + \partial_{H} \overset{ˇ}{w} {[\frac{\nabla b}{ε}]}^{⊤} + D_{\partial Σ} \overset{ˇ}{w} D_{x} σ, \\ \begin{matrix} Δ \vec{w} & = \partial_{ρ} \overset{ˇ}{w} [\frac{Δ r}{ε} - (Δ s \cdot \nabla_{Σ} {\overset{ˇ}{h}}_{ε} + \sum_{i, l = 1}^{N} \nabla s_{i} \cdot \nabla s_{l} {(\nabla_{Σ})}_{i} {(\nabla_{Σ})}_{l} {\overset{ˇ}{h}}_{ε})] + \partial_{H} \overset{ˇ}{w} \frac{Δ b}{ε} \\ + \partial_{H}^{2} \overset{ˇ}{w} \frac{| \nabla b |^{2}}{ε^{2}} + \partial_{ρ}^{2} \overset{ˇ}{w} {| \frac{\nabla r}{ε} - {(D_{x} s)}^{⊤} \nabla_{Σ} {\overset{ˇ}{h}}_{ε} |}^{2} + 2 \partial_{ρ} \partial_{H} \overset{ˇ}{w} \frac{\nabla b}{ε} \cdot [\frac{\nabla r}{ε} - {(D_{x} s)}^{⊤} \nabla_{Σ} {\overset{ˇ}{h}}_{ε}] \\ + 2 D_{\partial Σ} \partial_{ρ} \overset{ˇ}{w} D_{x} σ [\frac{\nabla r}{ε} - {(D_{x} s)}^{⊤} \nabla_{Σ} {\overset{ˇ}{h}}_{ε}] + 2 D_{\partial Σ} \partial_{H} \overset{ˇ}{w} D_{x} σ \frac{\nabla b}{ε} \\ + D_{\partial Σ} \overset{ˇ}{w} Δ σ + \sum_{i, l = 1}^{N} \nabla σ_{i} \cdot \nabla σ_{l} {(\nabla_{\partial Σ})}_{i} {(\nabla_{\partial Σ})}_{l} \overset{ˇ}{w}, \end{matrix} \end{array}$ where the $\vec{w}$ -terms on the left hand side and derivatives of r or s are evaluated at $(x, t)$ , the ${\overset{ˇ}{h}}_{ε}$ -terms at $(s (x, t), t)$ and the $\overset{ˇ}{w}$ -terms at $({\overset{ˇ}{ρ}}_{ε} (x, t), H_{ε} (x, t), σ (x, t), t)$ .
Proof.
This can be shown by applying Lemma 5.6 to every component. □

Contact point expansion: The bulk equation. We expand the $\nabla W$ -part in (5.37): If the ${\vec{u}}_{j}^{I}, {\vec{u}}_{j}^{C}$ are bounded, the Taylor expansion yields on $\overline{Γ (2 δ)}$ $\begin{matrix} \nabla W ({\vec{u}}_{ε}^{I} + {\vec{u}}_{ε}^{C}) = \nabla W ({\vec{θ}}_{0}) + \sum_{ν \in N_{0}^{m}, | ν | = 1}^{M + 2} \frac{1}{ν!} \partial_{y}^{ν} \nabla W ({\vec{θ}}_{0}) {[\sum_{j = 1}^{M + 1} ε^{j} ({\vec{u}}_{j}^{I} + {\vec{u}}_{j}^{C})]}^{ν} + O (ε^{M + 3}) . \end{matrix}$ As in the scalar case one can combine the latter with the expansion for $\nabla W ({\vec{u}}_{ε}^{I})$ in (5.28) and use ${\vec{u}}_{1}^{I} = 0$ . This yields that the terms in the asymptotic expansion for $\nabla W ({\vec{u}}_{ε}^{I} + {\vec{u}}_{ε}^{C}) - \nabla W ({\vec{u}}_{ε}^{I})$ are for $k = 1, \dots, M + 1$ : $\begin{array}{l} O (1) : 0, \\ O (ε) : D^{2} W ({\vec{θ}}_{0}) {\vec{u}}_{1}^{C}, \\ \begin{matrix} O (ε^{k}) : D^{2} W ({\vec{θ}}_{0}) {\vec{u}}_{k}^{C} + [ & some polynomial in entries of ({\vec{u}}_{1}^{I}, \dots, {\vec{u}}_{k - 1}^{I}, {\vec{u}}_{1}^{C}, \dots, {\vec{u}}_{k - 1}^{C}) of \\ order ⩽ k, where the coefficients are multiples of \partial_{y}^{ν} \nabla W ({\vec{θ}}_{0}), \\ ν \in N_{0}^{m}, | ν | = 2, \dots, k and every term contains a {({\vec{u}}_{j}^{C})}_{n} -factor] . \end{matrix} \end{array}$ The other explicit terms in $\nabla W ({\vec{u}}_{ε}^{I} + {\vec{u}}_{ε}^{C}) - \nabla W ({\vec{u}}_{ε}^{I})$ are of order $O (ε^{M + 3})$ .

Functions of $(s, t)$ , $(ρ, s, t)$ and $(x, t)$ are expanded in the analogous way as in the scalar case, cf. (5.13)–(5.14) and the remarks there. We just replace $h_{j}, ρ_{ε}$ by ${\overset{ˇ}{h}}_{j}, {\overset{ˇ}{ρ}}_{ε}$ .

As in the scalar case we use some notation for the higher orders in the expansion:
Definition 5.15 (Notation for Contact Point Expansion of (vAC)).

We call $({\vec{θ}}_{0}, {\vec{u}}_{1}^{I})$ the zero-th order and $({\overset{ˇ}{h}}_{j}, {\vec{u}}_{j + 1}^{I}, {\vec{u}}_{j}^{C})$ the jth order for $j = 1, \dots, M$ .

Let $k \in {- 1, \dots, M + 2}$ and $β, γ > 0$ . Then ${\overset{ˇ}{R}}_{k, (β, γ)}^{C}$ denotes the set of smooth functions $\vec{R} : \overline{R_{+}^{2}} \times \partial Σ \times [0, T] \to R^{m}$ depending only on the jth orders for $0 ⩽ j ⩽ min {k, M}$ and such that uniformly in $(ρ, H, σ, t)$ : $\begin{matrix} | \partial_{ρ}^{i} \partial_{H}^{l} {(\nabla_{\partial Σ})}_{n_{1}} \dots {(\nabla_{\partial Σ})}_{n_{d}} \partial_{t}^{n} \vec{R} (ρ, H, σ, t) | = O (e^{- (β | ρ | + γ H)}) \end{matrix}$ for all $n_{1}, \dots, n_{d} \in {1, \dots, N}$ and $d, i, l, n \in N_{0}$ .

The set ${\overset{ˇ}{R}}_{k, (β)}^{C}$ is defined in an analogous way without the H-dependence.

In the following we expand (5.37) into ε-series with coefficients in $({\overset{ˇ}{ρ}}_{ε}, H_{ε}, σ, t)$ .

Bulk Equation: $O (ε^{- 1})$ . The lowest order $O (\frac{1}{ε})$ in (5.37) cancels if $\begin{matrix} (5.38) & [- Δ (σ, t) + D^{2} W ({\vec{θ}}_{0} (ρ))] {\overset{ˇ}{u}}_{1}^{C} (ρ, H, σ, t) = 0, \end{matrix}$ where $Δ (σ, t) : = \partial_{ρ}^{2} + | \nabla b |^{2} |_{{\overline{X}}_{0} (σ, t)} \partial_{H}^{2}$ and we used $\nabla r \cdot \nabla b |_{{\overline{X}}_{0} (σ, t)} = 0$ for all $(σ, t) \in \partial Σ \times [0, T]$ .

Bulk Equation: $O (ε^{k - 1})$ . For $k = 1, \dots, M - 1$ we assume that the jth order is constructed for all $j = 0, \dots, k$ , that it is smooth and that ${\overset{ˇ}{u}}_{j + 1}^{I} \in {\overset{ˇ}{R}}_{j, (β)}^{I}$ (bounded and all derivatives bounded is enough here) and ${\overset{ˇ}{u}}_{j}^{C} \in {\overset{ˇ}{R}}_{j, (β, γ)}^{C}$ for every $β \in (0, min {\overset{ˇ}{β} (γ), \sqrt{λ / 2}, {\overset{ˇ}{β}}_{0}})$ , $γ \in (0, \overset{ˇ}{γ})$ , where ${\overset{ˇ}{β}}_{0}$ is from Theorem 4.13 and $\overset{ˇ}{β}, \overset{ˇ}{γ}$ are as in Theorem 4.18. With analogous computations as in the scalar case, the $O (ε^{k - 1})$ -order in the expansion for the bulk equation (5.37) is zero if $\begin{matrix} (5.39) & [- Δ (σ, t) + D^{2} W ({\vec{θ}}_{0})] {\overset{ˇ}{u}}_{k + 1}^{C} = {\vec{G}}_{k} (ρ, H, σ, t), \end{matrix}$ where ${\vec{G}}_{k} \in {\overset{ˇ}{R}}_{k, (β, γ)}^{C}$ .

Contact point expansion: The Neumann boundary condition. As in the scalar case, the boundary conditions complementing (5.38)–(5.39) will be obtained from the expansion of the Neumann boundary condition (vAC2) for ${\vec{u}}_{ε} = {\vec{u}}_{ε}^{I} + {\vec{u}}_{ε}^{C}$ , i.e. $D_{x} ({\vec{u}}_{ε}^{I} + {\vec{u}}_{ε}^{C}) |_{\partial Q_{T}} N_{\partial Ω} = 0$ . Lemma 5.12 and Lemma 5.14 yield on $\overline{Γ^{C} (2 δ, 2 μ_{1})}$ $\begin{array}{l} D_{x} {\vec{u}}_{ε}^{I} |_{(x, t)} = \partial_{ρ} {\overset{ˇ}{u}}_{ε}^{I} |_{(ρ, s, t)} {[\frac{\nabla r |_{(x, t)}}{ε} - {(D_{x} s)}^{⊤} |_{(x, t)} \nabla_{Σ} h_{ε} |_{(s, t)}]}^{⊤} + D_{Σ} {\overset{ˇ}{u}}_{ε}^{I} |_{(ρ, s, t)} D_{x} s |_{(x, t)}, \\ \begin{matrix} D_{x} {\vec{u}}_{ε}^{C} |_{(x, t)} & = \partial_{ρ} {\overset{ˇ}{u}}_{ε}^{C} |_{(ρ, H, σ, t)} {[\frac{\nabla r |_{(x, t)}}{ε} - {(D_{x} s)}^{⊤} |_{(x, t)} \nabla_{Σ} h_{ε} |_{(s, t)}]}^{⊤} \\ + \partial_{H} {\overset{ˇ}{u}}_{ε}^{C} |_{(ρ, H, σ, t)} {[\frac{\nabla b |_{(x, t)}}{ε}]}^{⊤} + D_{\partial Σ} {\overset{ˇ}{u}}_{ε}^{C} |_{(ρ, H, σ, t)} D_{x} σ |_{(x, t)}, \end{matrix} \end{array}$ where $ρ = {\overset{ˇ}{ρ}}_{ε} (x, t), H = H_{ε} (x, t)$ , $s = s (x, t)$ and $σ = σ (x, t)$ . We consider the points $x = X (r, σ, t)$ for $(r, σ, t) \in [- 2 δ, 2 δ] \times \partial Σ \times [0, T]$ , in particular $H = 0$ and $s = σ$ .

For $g : \overline{Γ (2 δ)} \cap \partial Q_{T} \to R$ smooth we use an expansion as in the scalar case, cf. (5.17) and the remarks there. We just use ${\overset{ˇ}{h}}_{j}, {\overset{ˇ}{ρ}}_{ε}$ instead of $h_{j}, ρ_{ε}$ .

In the following we expand the Neumann boundary condition into ε-series with coefficients in $({\overset{ˇ}{ρ}}_{ε}, σ, t)$ up to the order $O (ε^{M - 1})$ .

Neumann Boundary Condition: $O (ε^{- 1})$ . At the lowest order $O (\frac{1}{ε})$ we have $(N_{\partial Ω} \cdot \nabla r) |_{{\overline{X}}_{0} (σ, t)} {\vec{θ}}_{0}^{'} (ρ) = 0$ . This is valid due to the 90°-contact angle condition.

Neumann Boundary Condition: $O (ε^{0})$ . The order $O (1)$ vanishes if $\begin{array}{l} (5.40) & (N_{\partial Ω} \cdot \nabla b) |_{{\overline{X}}_{0} (σ, t)} \partial_{H} {\overset{ˇ}{u}}_{1}^{C} |_{H = 0} (ρ, σ, t) = {\vec{g}}_{1} (ρ, σ, t), \\ \begin{matrix} {\vec{g}}_{1} (ρ, σ, t) : = & {\vec{θ}}_{0}^{'} (ρ) [{(D_{x} s N_{\partial Ω})}^{⊤} |_{{\overline{X}}_{0} (σ, t)} \nabla_{Σ} {\overset{ˇ}{h}}_{1} |_{(σ, t)} - \partial_{r} ((N_{\partial Ω} \cdot \nabla r) \circ \overline{X}) |_{(0, σ, t)} {\overset{ˇ}{h}}_{1} |_{(σ, t)}] \\ + {\overset{ˇ}{g}}_{0} (ρ, σ, t), \end{matrix} \end{array}$ where ${\overset{ˇ}{g}}_{0} (ρ, σ, t) : = - ρ {\vec{θ}}_{0}^{'} (ρ) \partial_{r} ((N_{\partial Ω} \cdot \nabla r) \circ \overline{X}) |_{(0, σ, t)}$ . For $j = 1, \dots, M$ let $\begin{matrix} (5.41) & {\underline{u}}_{j}^{C} : \overline{R_{+}^{2}} \times \partial Σ \times [0, T] \to R^{m} : (ρ, H, σ, t) \mapsto {\overset{ˇ}{u}}_{j}^{C} (ρ, | \nabla b | ({\overline{X}}_{0} (σ, t)) H, σ, t) . \end{matrix}$ Due to Theorem 3.2 it holds $| \nabla b |_{{\overline{X}}_{0} (σ, t)} | ⩾ c > 0$ and $| N_{\partial Ω} \cdot \nabla b |_{{\overline{X}}_{0} (σ, t)} | ⩾ c > 0$ for all $(σ, t) \in \partial Σ \times [0, T]$ . Therefore (5.38) and (5.40) for ${\overset{ˇ}{u}}_{1}^{C}$ are equivalent to $\begin{array}{l} (5.42) & [- Δ + D^{2} W ({\vec{θ}}_{0} (ρ))] {\underline{u}}_{1}^{C} = 0, \\ (5.43) & - \partial_{H} {\underline{u}}_{1}^{C} |_{H = 0} = (| \nabla b | / N_{\partial Ω} \cdot \nabla b) |_{{\overline{X}}_{0} (σ, t)} {\vec{g}}_{1} (ρ, σ, t) . \end{array}$ The solvability condition (4.18) corresponding to (5.42)–(5.43) is $\begin{matrix} (| \nabla b | / N_{\partial Ω} \cdot \nabla b) |_{{\overline{X}}_{0} (σ, t)} \int_{R} {\vec{g}}_{1} (ρ, σ, t) \cdot {\vec{θ}}_{0}^{'} (ρ) d ρ = 0 . \end{matrix}$ Due to the symmetry properties of ${\vec{θ}}_{0}$ , the term coming from ${\overset{ˇ}{g}}_{0}$ vanishes. Therefore the latter condition yields the following boundary condition for ${\overset{ˇ}{h}}_{1}$ : $\begin{matrix} (5.44) & b_{1} (σ, t) \cdot \nabla_{Σ} {\overset{ˇ}{h}}_{1} |_{(σ, t)} + b_{0} (σ, t) {\overset{ˇ}{h}}_{1} |_{(σ, t)} = 0 for (σ, t) \in \partial Σ \times [0, T], \end{matrix}$ where $b_{1}, b_{0}$ are as in the scalar case, cf. the formulas below (5.22). Together with the linear parabolic equation (5.31) for ${\overset{ˇ}{h}}_{1}$ from Section 5.2.1, we obtain a time-dependent linear parabolic boundary value problem for ${\overset{ˇ}{h}}_{1}$ , where the initial value ${\overset{ˇ}{h}}_{1} |_{t = 0}$ is not prescribed yet. However, since ${\overset{ˇ}{f}}_{0}$ is zero, the equations for ${\overset{ˇ}{h}}_{1}$ are homogeneous and we can take ${\overset{ˇ}{h}}_{1} = 0$ .

Hence we get ${\overset{ˇ}{u}}_{2}^{I}$ from Section 5.2.1 with ${\overset{ˇ}{u}}_{2}^{I} \in {\overset{ˇ}{R}}_{1, (β_{1})}^{I}$ for all $β_{1} \in (0, min {\sqrt{λ / 2}, {\overset{ˇ}{β}}_{0}})$ , where ${\overset{ˇ}{β}}_{0} > 0$ is as in Theorem 4.13. In particular the first inner order is determined. Furthermore, we have ${\vec{g}}_{1} \in {\tilde{R}}_{1, (β_{1})}^{I}$ for all $β_{1} \in (0, \sqrt{λ / 2})$ due to Theorem 4.9. With Theorem 4.18 (applied in local coordinates for $\partial Σ$ ) there is a unique smooth solution ${\underline{u}}_{1}^{C}$ to (5.42)–(5.43) and we get decay properties. By compactness and Remark 5.4 with $\partial Σ$ instead of Σ we obtain the decay ${\underline{u}}_{1}^{C} \in {\overset{ˇ}{R}}_{1, (β, γ)}^{C}$ for all $β \in (0, min {\overset{ˇ}{β} (γ), \sqrt{λ / 2}})$ , $γ \in (0, \overset{ˇ}{γ})$ , where $\overset{ˇ}{β}, \overset{ˇ}{γ}$ are as in Theorem 4.18. Altogether we computed the first order.

Neumann Boundary Condition: $O (ε^{k})$ and Induction. For $k = 1, \dots, M - 1$ we compute $O (ε^{k})$ in (vAC2) for ${\vec{u}}_{ε} = {\vec{u}}_{ε}^{I} + {\vec{u}}_{ε}^{C}$ and obtain equations for the ( $k + 1$ )-th order. We use the following induction hypothesis: assume that the jth order is constructed for all $j = 0, \dots, k$ , that it is smooth and has the decay ${\overset{ˇ}{u}}_{j + 1}^{I} \in {\overset{ˇ}{R}}_{j, (β_{1})}^{I}$ for all $β_{1} \in (0, min {\sqrt{λ / 2}, {\overset{ˇ}{β}}_{0}})$ as well as ${\overset{ˇ}{u}}_{j}^{C} \in {\overset{ˇ}{R}}_{j, (β, γ)}^{C}$ for all $β \in (0, min {\overset{ˇ}{β} (γ), \sqrt{λ / 2}, {\overset{ˇ}{β}}_{0}})$ , $γ \in (0, \overset{ˇ}{γ})$ , where ${\overset{ˇ}{β}}_{0}$ is from Theorem 4.13 and $\overset{ˇ}{β}, \overset{ˇ}{γ}$ are as in Theorem 4.18. The assumption holds for $k = 1$ due to the last subparagraph.

Analogously as in the scalar case, the $O (ε^{k})$ -order in (vAC2) for ${\vec{u}}_{ε} = {\vec{u}}_{ε}^{I} + {\vec{u}}_{ε}^{C}$ is zero if $\begin{array}{l} (5.45) & (N_{\partial Ω} \cdot \nabla b) |_{{\overline{X}}_{0} (σ, t)} \partial_{H} {\overset{ˇ}{u}}_{k + 1}^{C} |_{H = 0} (ρ, σ, t) = {\vec{g}}_{k + 1} (ρ, σ, t), \\ \begin{matrix} {\vec{g}}_{k + 1} |_{(ρ, σ, t)} : = & {\vec{θ}}_{0}^{'} (ρ) [{(D_{x} s N_{\partial Ω})}^{⊤} |_{{\overline{X}}_{0} (σ, t)} \nabla_{Σ} {\overset{ˇ}{h}}_{k + 1} |_{(σ, t)} - \partial_{r} ((N_{\partial Ω} \cdot \nabla r) \circ \overline{X}) |_{(0, σ, t)} {\overset{ˇ}{h}}_{k + 1} |_{(σ, t)}] \\ + {\overset{ˇ}{g}}_{k} (ρ, σ, t), \end{matrix} \end{array}$ where ${\overset{ˇ}{g}}_{k} \in {\overset{ˇ}{R}}_{k, (β)}^{C}$ and hence ${\vec{g}}_{k + 1} \in {\tilde{R}}_{k + 1, (β)}^{I} + {\overset{ˇ}{R}}_{k, (β)}^{C}$ , if ${\overset{ˇ}{h}}_{k + 1}$ is smooth.

As in the last subparagraph, the equations (5.39), (5.45) are equivalent to $\begin{array}{l} (5.46) & [- Δ + D^{2} W ({\vec{θ}}_{0} (ρ))] {\underline{u}}_{k + 1}^{C} = {\underline{G}}_{k} (ρ, H, σ, t), \\ (5.47) & - \partial_{H} {\underline{u}}_{k + 1}^{C} |_{H = 0} = (| \nabla b | / N_{\partial Ω} \cdot \nabla b) |_{{\overline{X}}_{0} (σ, t)} {\vec{g}}_{k + 1} (ρ, H, σ, t), \end{array}$ where we defined ${\underline{u}}_{k + 1}^{C}$ in (5.41) and ${\underline{G}}_{k}$ is defined in the analogous way with the ${\vec{G}}_{k} \in {\overset{ˇ}{R}}_{k, (β, γ)}^{C}$ from (5.39). The compatibility condition (4.18) for (5.46)–(5.47), i.e. $\begin{matrix} \int_{R_{+}^{2}} {\underline{G}}_{k} (ρ, H, σ, t) \cdot {\vec{θ}}_{0}^{'} (ρ) d (ρ, H) + (| \nabla b | / N_{\partial Ω} \cdot \nabla b) |_{{\overline{X}}_{0} (σ, t)} \int_{R} {\vec{g}}_{k + 1} (ρ, σ, t) \cdot {\vec{θ}}_{0}^{'} (ρ) d ρ = 0, \end{matrix}$ implies a linear boundary condition for ${\overset{ˇ}{h}}_{k + 1}$ : $\begin{matrix} (5.48) & b_{1} (σ, t) \cdot \nabla_{Σ} {\overset{ˇ}{h}}_{k + 1} |_{(σ, t)} + b_{0} (σ, t) {\overset{ˇ}{h}}_{k + 1} |_{(σ, t)} = {\overset{ˇ}{f}}_{k}^{B} (σ, t) for (σ, t) \in \partial Σ \times [0, T], \end{matrix}$ where $b_{0}, b_{1}$ are defined below (5.22) and $\begin{matrix} {\overset{ˇ}{f}}_{k}^{B} |_{(σ, t)} : = \frac{- 1}{‖ {\vec{θ}}_{0}^{'} ‖_{L^{2} {(R)}^{m}}^{2}} [\int_{R_{+}^{2}} {\underline{G}}_{k} |_{(ρ, H, σ, t)} \cdot {\vec{θ}}_{0}^{'} |_{ρ} d ρ + \frac{| \nabla b |}{N_{\partial Ω} \cdot \nabla b} |_{{\overline{X}}_{0} (σ, t)} \int_{R} {\overset{ˇ}{g}}_{k} |_{(ρ, σ, t)} \cdot {\vec{θ}}_{0}^{'} |_{ρ} d ρ] \end{matrix}$ is smooth in $(σ, t) \in \partial Σ \times [0, T]$ .

Because of the remarks and computations in Section 5.1.2 we can solve (5.36) from Section 5.2.1 together with (5.48) and obtain a smooth solution ${\overset{ˇ}{h}}_{k + 1}$ . Therefore Section 5.2.1 yields ${\overset{ˇ}{u}}_{k + 2}^{I}$ (solving (5.35)) with ${\overset{ˇ}{u}}_{k + 2}^{I} \in {\overset{ˇ}{R}}_{k + 1, (β_{1})}^{I}$ for all $β_{1} \in (0, min {\sqrt{λ / 2}, {\overset{ˇ}{β}}_{0}})$ . In particular the $(k + 1)$ th inner order is computed and it holds ${\vec{G}}_{k} \in {\overset{ˇ}{R}}_{k, (β, γ)}^{C}$ as well as ${\vec{g}}_{k + 1} \in {\tilde{R}}_{k + 1, (β)}^{I} + {\overset{ˇ}{R}}_{k, (β)}^{C}$ for all $β \in (0, min {\overset{ˇ}{β} (γ), \sqrt{λ / 2}, {\overset{ˇ}{β}}_{0}})$ , $γ \in (0, \overset{ˇ}{γ})$ . As in the last subparagraph we obtain a unique smooth solution ${\underline{u}}_{k + 1}^{C}$ to (5.46)–(5.47) with the decay ${\overset{ˇ}{u}}_{k + 1}^{C} \in {\overset{ˇ}{R}}_{k, (β, γ)}^{C}$ for all $(β, γ)$ as above. Altogether, the $(k + 1)$ th order is determined.

Finally, by induction the jth order is constructed for all $j = 0, \dots, M$ , the ${\overset{ˇ}{h}}_{j}$ are smooth and ${\overset{ˇ}{u}}_{j + 1}^{I} \in {\overset{ˇ}{R}}_{j, (β_{1})}^{I}$ for all for all $β_{1} \in (0, min {\sqrt{λ / 2}, {\overset{ˇ}{β}}_{0}})$ as well as ${\overset{ˇ}{u}}_{j}^{C} \in {\overset{ˇ}{R}}_{j, (β, γ)}^{C}$ for every $β \in (0, min {\overset{ˇ}{β} (γ), \sqrt{λ / 2}, {\overset{ˇ}{β}}_{0}})$ , $γ \in (0, \overset{ˇ}{γ})$ .

5.2.3. The approximate solution for (vAC) in ND

Let $N ⩾ 2$ and $Γ = {(Γ_{t})}_{t \in [0, T]}$ be as in Section 3.1 with contact angle $\frac{π}{2}$ and a solution to (MCF) in Ω. Moreover, let $δ > 0$ be such that the assertions of Theorem 3.2 hold for $2 δ$ instead of δ and let $r, s, b, σ, μ_{0}$ be as in the theorem. Furthermore, let $W : R^{m} \to R$ be as in Definition 1.1 and ${\vec{u}}_{\pm}$ be any distinct pair of minimizers of W. Moreover, let $M \in N$ , $M ⩾ 2$ be as in the beginning of Section 5.2. Let $η : R \to [0, 1]$ be smooth with $η (r) = 1$ for $| r | ⩽ 1$ and $η (r) = 0$ for $| r | ⩾ 2$ . Then for $ε > 0$ we set $\begin{matrix} {\vec{u}}_{ε}^{A} : = \{\begin{matrix} η (\frac{r}{δ}) [{\vec{u}}_{ε}^{I} + {\vec{u}}_{ε}^{C} η (\frac{b}{μ_{1}})] + (1 - η (\frac{r}{δ})) {\vec{u}}_{sign (r)} & in \overline{Γ (2 δ)}, \\ {\vec{u}}_{\pm} & in Q_{T}^{\pm} ∖ Γ (2 δ), \end{matrix} \end{matrix}$ where ${\vec{u}}_{ε}^{I}$ and ${\vec{u}}_{ε}^{C}$ were constructed in Sections 5.2.1 and 5.2.2, and for convenience we have set ${\vec{u}}_{\pm 1} : = {\vec{u}}_{\pm}$ . Analogously as in Section 5.1.3 one can prove that ${\vec{u}}_{ε}^{A}$ is an approximate solution for (vAC1)–(vAC3) in the following sense:

Lemma 5.16.
The function ${\vec{u}}_{ε}^{A}$ is smooth, uniformly bounded with respect to $x, t, ε$ and for the remainder ${\vec{r}}_{ε}^{A} : = \partial_{t} {\vec{u}}_{ε}^{A} - Δ {\vec{u}}_{ε}^{A} + \frac{1}{ε^{2}} \nabla W ({\vec{u}}_{ε}^{A})$ in ( vAC1 ) and ${\vec{s}}_{ε}^{A} : = \partial_{N_{\partial Ω}} {\vec{u}}_{ε}^{A}$ in ( vAC2 ) it holds $\begin{array}{l} | {\vec{r}}_{ε}^{A} | ⩽ C (ε^{M} e^{- c | {\overset{ˇ}{ρ}}_{ε} |} + ε^{M + 1}) in Γ (2 δ, μ_{1}), \\ | {\vec{r}}_{ε}^{A} | ⩽ C (ε^{M - 1} e^{- c (| {\overset{ˇ}{ρ}}_{ε} | + H_{ε})} + ε^{M} e^{- c | {\overset{ˇ}{ρ}}_{ε} |} + ε^{M + 1}) in Γ^{C} (2 δ, 2 μ_{1}), \\ {\vec{r}}_{ε}^{A} = 0 in Q_{T} ∖ Γ (2 δ), \\ | {\vec{s}}_{ε}^{A} | ⩽ C ε^{M} e^{- c | {\overset{ˇ}{ρ}}_{ε} |} on \partial Q_{T} \cap Γ (2 δ), \\ {\vec{s}}_{ε}^{A} = 0 on \partial Q_{T} ∖ Γ (2 δ) \end{array}$ for $ε > 0$ small and some $c, C > 0$ . Here ${\overset{ˇ}{ρ}}_{ε}$ is defined in ( 5.27 ) and $H_{ε} = \frac{b}{ε}$ .
Remark 5.17.
The analogous assertions as in Remark 5.11 hold.

6. Difference estimates and proofs of the convergence theorems

In this section we estimate the difference of the exact and approximate solutions with a Gronwall-type argument in both cases. This is the second step in the method by de Mottoni and Schatzman [30]. Here the major ingredients are the spectral estimates from [67]. Moreover, we have to control certain nonlinear terms stemming from differences of potential terms. We will estimate these with interpolation inequalities. Therefore as preparation we prove uniform a priori bounds for exact classical solutions of (AC) and (vAC) in Sections 6.1.1 and 6.1.2, respectively. Moreover we recall some Gagliardo–Nirenberg estimates in Section 6.1.3. For Allen–Cahn type models such uniform bounds in ε for the exact solutions is typical, see de Mottoni, Schatzman [30, Section 6] for the standard Allen–Cahn equation as well as [2, Remark 1.2 and Section 5.2] for the Allen–Cahn equation coupled with the Stokes system. In Section 6.2 we prove the difference estimate in the scalar case and Theorem 1.4. This is similar to [5, Sections 5–6], but more technical. Finally, in Section 6.3 we show the difference estimate in the vector-valued case and Theorem 1.6. The latter works analogously to the scalar case.

6.1. Preliminaries

6.1.1. Uniform a priori bound for classical solutions of (AC)

Let N, Ω, $Q_{T}$ , $\partial Q_{T}$ be as in Remark 1.3, 1. and $ε > 0$ . We prove uniform boundedness estimates for classical solutions of the Allen–Cahn equation (AC). Let f be as in (1.1) and $R_{0} ⩾ 1$ be such that the condition (1.2) for $f^{'}$ holds.

Lemma 6.1.
Let $u_{0, ε} \in C^{0} (\overline{Ω})$ and $u_{ε} \in C^{0} (\overline{Q_{T}}) \cap C^{1} (\overline{Ω} \times (0, T]) \cap C^{2} (Ω \times (0, T])$ be a solution of ( AC1 )–( AC3 ). Then $\begin{matrix} ‖ u_{ε} ‖_{L^{\infty} (Q_{T})} ⩽ max {R_{0}, ‖ u_{0, ε} ‖_{L^{\infty} (Ω)}} . \end{matrix}$
Proof.
The proof is done via contradiction with ideas from the maximum principle. The argument works similarly in the vector-valued case which is slightly more complicated, see Lemma 6.2 below. Therefore we leave out the proof. See [66, Lemma 7.1] for the details. □

6.1.2. Uniform a priori bound for classical solutions of (vAC)

Let N, Ω, $Q_{T}$ , $\partial Q_{T}$ be as in Remark 1.3, 1. and $ε > 0$ . Moreover, let $m \in N$ and $W : R^{m} \to R$ be as in Definition 1.1. We prove uniform boundedness estimates for classical solutions of (vAC). This works analogously to the proof of Lemma 6.1 for the scalar case in the last section.

Lemma 6.2.
Let ${\vec{u}}_{0, ε} \in C^{0} {(\overline{Ω})}^{m}$ and ${\vec{u}}_{ε} \in C^{0} {(\overline{Q_{T}})}^{m} \cap C^{1} (\overline{Ω} \times {(0, T])}^{m} \cap C^{2} (Ω \times {(0, T])}^{m}$ be a solution of ( vAC1 )–( vAC3 ). Then with ${\overset{ˇ}{R}}_{0} > 0$ as in Definition 1.1 it holds $\begin{matrix} ‖ {\vec{u}}_{ε} ‖_{L^{\infty} (Q_{T}, R^{m})} ⩽ max {{\overset{ˇ}{R}}_{0}, ‖ {\vec{u}}_{0, ε} ‖_{L^{\infty} (Ω, R^{m})}} . \end{matrix}$
Proof.
We use a contradiction argument and ideas from the proof of the weak maximum principle for parabolic equations, cf. Renardy, Rogers [71, Theorem 4.25]. We make the assumption that $‖ {\vec{u}}_{ε} ‖_{L^{\infty} (Q_{T}, R^{m})} > max {{\overset{ˇ}{R}}_{0}, ‖ {\vec{u}}_{0, ε} ‖_{L^{\infty} (Ω, R^{m})}}$ . Let ${\overset{ˇ}{u}}_{ε, β} : = | {\vec{u}}_{ε} |^{2} + β e^{- t}$ and ${\overset{ˇ}{u}}_{0, ε, β} : = | {\vec{u}}_{0, ε} |^{2} + β$ for $β > 0$ . Then for $β > 0$ small $\begin{matrix} (6.1) & ‖ {\overset{ˇ}{u}}_{ε, β} ‖_{L^{\infty} (Q_{T})} > β + max {{\overset{ˇ}{R}}_{0}^{2}, ‖ {\overset{ˇ}{u}}_{0, ε, β} ‖_{L^{\infty} (Ω)}} . \end{matrix}$ The maximum of ${\overset{ˇ}{u}}_{ε, β} = | {\overset{ˇ}{u}}_{ε, β} |$ is attained in some $(x_{0}, t_{0}) \in \overline{Q_{T}}$ due to ${\vec{u}}_{ε} \in C^{0} {(\overline{Q_{T}})}^{m}$ . Then inequality (6.1) yields $\begin{matrix} (6.2) & | {\vec{u}}_{ε} |^{2} |_{(x_{0}, t_{0})} = {\overset{ˇ}{u}}_{ε, β} |_{(x_{0}, t_{0})} - β e^{- t_{0}} > β + {\overset{ˇ}{R}}_{0}^{2} - β = {\overset{ˇ}{R}}_{0}^{2} . \end{matrix}$ By the assumption on W in Definition 1.1 it follows that ${\vec{u}}_{ε} |_{(x_{0}, t_{0})} \cdot \nabla W ({\vec{u}}_{ε} |_{(x_{0}, t_{0})}) ⩾ 0$ . Hence equation (vAC1) yields in the case $(x_{0}, t_{0}) \in Ω \times (0, T]$ that $\begin{array}{l} (\partial_{t} - Δ) {\overset{ˇ}{u}}_{ε, β} |_{(x_{0}, t_{0})} & = - β e^{- t_{0}} + 2 {\vec{u}}_{ε} \cdot (\partial_{t} {\vec{u}}_{ε} - Δ {\vec{u}}_{ε}) |_{(x_{0}, t_{0})} - 2 | \nabla {\vec{u}}_{ε} |^{2} |_{(x_{0}, t_{0})} \\ = - β e^{- t_{0}} - 2 [{\vec{u}}_{ε} |_{(x_{0}, t_{0})} \cdot \frac{1}{ε^{2}} \nabla W ({\vec{u}}_{ε} |_{(x_{0}, t_{0})}) + | \nabla {\vec{u}}_{ε} |^{2} |_{(x_{0}, t_{0})}] \\ ⩽ - β e^{- t_{0}} < 0 . \end{array}$ In the case $(x_{0}, t_{0}) \in \overline{Ω} \times {0}$ the contradiction follows from (6.1) and ${\overset{ˇ}{u}}_{ε, β} |_{t = 0} = {\overset{ˇ}{u}}_{0, ε, β}$ because of (vAC3). In the case $(x_{0}, t_{0}) \in Ω \times (0, T]$ we obtain a contradiction to $(\partial_{t} - Δ) {\overset{ˇ}{u}}_{ε, β} |_{(x_{0}, t_{0})} ⩾ 0$ as in the proof of the weak maximum principle, cf. [71, Theorem 4.25].

Finally, let $(x_{0}, t_{0}) \in \partial Ω \times (0, T]$ . The above consideration yields ${\overset{ˇ}{u}}_{ε, β} |_{(x_{0}, t_{0})} > {\overset{ˇ}{u}}_{ε, β} |_{(x, t)}$ for all $(x, t) \in Ω \times (0, T]$ . Moreover, due to continuity and (6.2) we obtain $| {\vec{u}}_{ε} (x, t) | > {\overset{ˇ}{R}}_{0}$ for all $(x, t) \in B_{η} (x_{0}, t_{0}) \cap (Ω \times (0, T])$ and $η > 0$ small. Therefore as above it follows that $(\partial_{t} - Δ) {\overset{ˇ}{u}}_{ε, β} |_{(x, t)} ⩽ - β e^{- t} < 0$ for these $(x, t)$ . Furthermore, it holds $\partial_{t} {\overset{ˇ}{u}}_{ε, β} |_{(x_{0}, t_{0})} ⩾ 0$ because $(x_{0}, t_{0})$ is a maximum of ${\overset{ˇ}{u}}_{ε, β}$ . By continuity of $\partial_{t} {\overset{ˇ}{u}}_{ε, β}$ we obtain $Δ {\overset{ˇ}{u}}_{ε, β} |_{(x, t)} < 0$ for all $(x, t) \in B_{η} (x_{0}, t_{0}) \cap (Ω \times (0, T])$ and $η > 0$ small. Therefore the Hopf Lemma (cf. [44, Lemma 3.4]) can be applied on $B_{η} (x_{0}) \cap Ω$ and yields $N_{\partial Ω} \cdot \nabla {\overset{ˇ}{u}}_{ε, β} |_{(x_{0}, t_{0})} > 0$ . Here $\nabla {\overset{ˇ}{u}}_{ε, β} = \sum_{j = 1}^{m} \nabla (| u_{ε, j} |^{2}) = 2 \sum_{j = 1}^{m} u_{ε, j} \nabla u_{ε, j}$ . Therefore we obtain a contradiction to the boundary condition (vAC2). Finally, this yields the lemma. □

6.1.3. Gagliardo–Nirenberg inequalities

Let us recall some Gagliardo–Nirenberg inequalities.

Lemma 6.3 (Gagliardo–Nirenberg Inequality).

Let $n \in N$ , $1 ⩽ p, q, r ⩽ \infty$ and $θ \in [0, 1]$ such that $\begin{matrix} θ (\frac{1}{p} - \frac{1}{n}) + \frac{1 - θ}{q} = \frac{1}{r}, \end{matrix}$ where $\frac{1}{\infty} : = 0$ . Moreover, if $p = n > 1$ , then we assume $θ < 1$ . Then $\begin{matrix} ‖ u ‖_{L^{r} (R^{n})} ⩽ c ‖ u ‖_{L^{q} (R^{n})}^{1 - θ} ‖ \nabla u ‖_{L^{p} (R^{n})}^{θ} \end{matrix}$ for all $u \in L^{q} (R^{n}) \cap W^{1, p} (R^{n})$ and a constant $c = c (n, p, q, r) > 0$ .

Proof.
See Leoni [58, Theorem 12.83]. □
Remark 6.4.
With suitable extension operators the estimate in Lemma 6.3 carries over to domains Ω with uniform Lipschitz boundary if the $\nabla u$ -factor in the estimate is replaced by the $W^{1, p}$ -norm and the constant in the estimate depends on n, p, q, r and the operator norm of the extension operator. For the existence of such extension operators (going back to Stein) see Leoni [58, Theorems 13.8, 13.17]. Note that the operator norms in [58] are estimated solely in terms of the usual parameters and the geometrical quantities of Ω and $\partial Ω$ . In particular if the geometrical quantities can be controlled in a uniform way, the operator norms and the constants in the above Gagliardo–Nirenberg inequalities can be taken uniformly with respect to Ω.

6.2. Difference estimate and proof of the convergence theorem for (AC) in ND

We prove in Section 6.2.1 a rather abstract estimate for the difference of exact solutions and suitable approximate solutions for the Allen–Cahn equation (AC1)–(AC3) in ND. Then in Section 6.2.2 we show the Theorem 1.4 about convergence by verifying the requirements for the difference estimate applied to the approximate solution from Section 5.1.3.

6.2.1. Difference estimate

Theorem 6.5 (Difference Estimate for (AC)).

Let $N ⩾ 2$ , Ω, $Q_{T}$ and $\partial Q_{T}$ be as in Remark 1.3 , 1. Moreover, let $Γ = {(Γ_{t})}_{t \in [0, T_{0}]}$ for some $T_{0} > 0$ be as in Section 3.2 and $δ > 0$ be such that Theorem 3.2 holds for $2 δ$ instead of δ. We use the notation for $Γ_{t} (δ)$ , $Γ (δ)$ , $\nabla_{τ}$ and $\partial_{n}$ from Remark 3.3 . Additionally, let f satisfy ( 1.1 )–( 1.2 ).

Moreover, let $ε_{0} > 0$ , $u_{ε}^{A} \in C^{2} (\overline{Q_{T_{0}}})$ , $u_{0, ε} \in C^{2} (\overline{Ω})$ with $\partial_{N_{\partial Ω}} u_{0, ε} = 0$ on $\partial Ω$ and let $u_{ε} \in C^{2} (\overline{Q_{T_{0}}})$ be exact solutions to ( AC1 )–( AC3 ) with $u_{0, ε}$ in ( AC3 ) for $ε \in (0, ε_{0}]$ .

For some $R > 0$ and $M \in N, M ⩾ k (N) : = max {2, \frac{N}{2}}$ we impose the following conditions:

Uniform Boundedness: ${sup}_{ε \in (0, ε_{0}]} ‖ u_{ε}^{A} ‖_{L^{\infty} (Q_{T_{0}})} + ‖ u_{0, ε} ‖_{L^{\infty} (Ω)} < \infty$ .

Spectral Estimate: There are $c_{0}, C > 0$ such that $\begin{matrix} \int_{Ω} | \nabla ψ |^{2} + \frac{1}{ε^{2}} f^{″} (u_{ε}^{A} (\cdot, t)) ψ^{2} d x ⩾ - C ‖ ψ ‖_{L^{2} (Ω)}^{2} + ‖ \nabla ψ ‖_{L^{2} (Ω ∖ Γ_{t} (δ))}^{2} + c_{0} ‖ \nabla_{τ} ψ ‖_{L^{2} (Γ_{t} (δ))}^{2} \end{matrix}$ for all $ψ \in H^{1} (Ω)$ and $ε \in (0, ε_{0}], t \in [0, T_{0}]$ .

Approximate Solution: For the remainders $\begin{matrix} r_{ε}^{A} : = \partial_{t} u_{ε}^{A} - Δ u_{ε}^{A} + \frac{1}{ε^{2}} f^{'} (u_{ε}^{A}) and s_{ε}^{A} : = \partial_{N_{\partial Ω}} u_{ε}^{A} \end{matrix}$ in ( AC1 )–( AC2 ) for $u_{ε}^{A}$ and the difference ${\overline{u}}_{ε} : = u_{ε} - u_{ε}^{A}$ it holds $\begin{matrix} (6.3) & \begin{matrix} | \int_{\partial Ω} s_{ε}^{A} tr {\overline{u}}_{ε} (t) d H^{N - 1} + \int_{Ω} r_{ε}^{A} {\overline{u}}_{ε} (t) d x | \\ ⩽ C ε^{M + \frac{1}{2}} ({‖ {\overline{u}}_{ε} (t) ‖}_{L^{2} (Ω)} + {‖ \nabla_{τ} {\overline{u}}_{ε} (t) ‖}_{L^{2} (Γ_{t} (δ))} + {‖ \nabla {\overline{u}}_{ε} (t) ‖}_{L^{2} (Ω ∖ Γ_{t} (δ))}) \end{matrix} \end{matrix}$ for all $ε \in (0, ε_{0}]$ and $t \in (0, T_{0}]$ .

Well-Prepared Initial Data: For all $ε \in (0, ε_{0}]$ it holds $\begin{matrix} (6.4) & {‖ u_{0, ε} - u_{ε}^{A} |_{t = 0} ‖}_{L^{2} (Ω)} ⩽ R ε^{M + \frac{1}{2}} . \end{matrix}$

Then we obtain

Let $M > k (N)$ . Then there are $β, ε_{1} > 0$ such that for $g_{β} (t) : = e^{- β t}$ it holds $\begin{matrix} (6.5) & \begin{matrix} sup_{t \in [0, T]} {‖ (g_{β} {\overline{u}}_{ε}) (t) ‖}_{L^{2} (Ω)}^{2} + ‖ g_{β} \nabla {\overline{u}}_{ε} ‖_{L^{2} (Q_{T} ∖ Γ (δ))}^{2} ⩽ 2 R^{2} ε^{2 M + 1}, \\ c_{0} ‖ g_{β} \nabla_{τ} {\overline{u}}_{ε} ‖_{L^{2} (Q_{T} \cap Γ (δ))}^{2} + ε^{2} ‖ g_{β} \partial_{n} {\overline{u}}_{ε} ‖_{L^{2} (Q_{T} \cap Γ (δ))}^{2} ⩽ 2 R^{2} ε^{2 M + 1} \end{matrix} \end{matrix}$ for all $ε \in (0, ε_{1}]$ and $T \in (0, T_{0}]$ .

Let $k (N) \in N$ and $M = k (N)$ . Let ( 6.3 ) hold for some $\tilde{M} > M$ instead of M. Then there are $β, \tilde{R}, ε_{1} > 0$ such that, if ( 6.4 ) holds for $\tilde{R}$ instead of R, then ( 6.5 ) for $\tilde{R}$ instead of R is valid for all $ε \in (0, ε_{1}], T \in (0, T_{0}]$ .

Let $N \in {2, 3}$ and $M = 2 (= k (N))$ . Then there are $ε_{1}, T_{1} > 0$ such that ( 6.5 ) holds for $β = 0$ and for all $ε \in (0, ε_{1}], T \in (0, T_{1}]$ .

Remark 6.6.

The parameter M corresponds to the order of the approximate solution in Section 5.1.

The parameter β was introduced in order to obtain a result valid for all times $T \in (0, T_{0}]$ .

Note that weaker requirements in the theorem also work, e.g. when one does not have the two additional terms on the right hand side of the spectral estimate or only an estimate with the full $H^{1}$ -norm on the right hand side in (6.3). Moreover, a slightly less involved proof is also possible, see e.g. Remark 6.8 below. However, then the result is also weaker, in particular the somewhat critical order $k (N)$ for M could be increased and the ε-orders in (6.5) could be weakened. This is because the $H^{1}$ -norm can be controlled with the spectral term but one has to pay $ε^{- 2}$ times the $L^{2}$ -norm. Nevertheless, we intended to give an optimal result, also having in mind e.g. couplings with other equations like the Stokes system as in [2], where a low number of terms in the ansatz for the approximate solution is convenient.

That the parameter $k (N)$ is critical for M in our proof can be seen at (6.11) in the proof below. This is due to an estimate of a cubic term, see (6.10) and Lemma 6.7 below. The results 2.–3. are the best we could prove for the critical case $M = k (N) \in N$ . This situation is difficult because in the estimates there will be a term of order larger than 2 in R and a linear term in R, but the desired order is 2 in R. The linear term in R will enter due to (6.3), see (6.11) below. For the parameter β there is a similar problem in the critical case.

Proof of Theorem 6.5.
The continuity of the objects on the left hand side in (6.5) yields that $\begin{matrix} (6.6) & T_{ε, β, R} : = sup {\tilde{T} \in (0, T_{0}] : (6.5) holds for ε, R and all T \in (0, \tilde{T}]} \end{matrix}$ is well-defined for all $ε \in (0, ε_{0}], β ⩾ 0$ and $T_{ε, β, R} > 0$ . In the different cases we have to show:
If $M > k (N)$ , then there are $β, ε_{1} > 0$ such that $T_{ε, β, R} = T_{0}$ for all $ε \in (0, ε_{1}]$ .

If $k (N) \in N$ and $M = k (N)$ , then there are $β, \tilde{R}, ε_{1} > 0$ such that $T_{ε, β, \tilde{R}} = T_{0}$ provided that $ε \in (0, ε_{1}]$ and (6.3) is true for some $\tilde{M} > M$ instead of M and (6.4) is valid with R replaced by $\tilde{R}$ .

If $N \in {2, 3}$ , $M = 2$ , then there are $T_{1}, ε_{1} > 0$ such that $T_{ε, 0, R} ⩾ T_{1}$ for all $ε \in (0, ε_{1}]$ .

We carry out a general computation first and return back to the different cases later. The difference of the left hand sides in (AC1) for $u_{ε}$ and $u_{ε}^{A}$ yields $\begin{matrix} (6.7) & [\partial_{t} - Δ + \frac{1}{ε^{2}} f^{″} (u_{ε}^{A})] {\overline{u}}_{ε} = - r_{ε}^{A} - r_{ε} (u_{ε}, u_{ε}^{A}), \end{matrix}$ where $r_{ε} (u_{ε}, u_{ε}^{A}) : = \frac{1}{ε^{2}} [f^{'} (u_{ε}) - f^{'} (u_{ε}^{A}) - f^{″} (u_{ε}^{A}) {\overline{u}}_{ε}]$ . We multiply (6.7) by $g_{β}^{2} {\overline{u}}_{ε}$ and integrate over $Q_{T}$ for $T \in (0, T_{ε, β, R}]$ , where $ε \in (0, ε_{0}]$ and $β ⩾ 0$ are fixed. This yields $\begin{matrix} (6.8) & \int_{0}^{T} g_{β}^{2} \int_{Ω} {\overline{u}}_{ε} [\partial_{t} - Δ + \frac{1}{ε^{2}} f^{″} (u_{ε}^{A})] {\overline{u}}_{ε} d x d t = - \int_{0}^{T} g_{β}^{2} \int_{Ω} [r_{ε}^{A} + r_{ε} (u_{ε}, u_{ε}^{A})] {\overline{u}}_{ε} d x d t \end{matrix}$ for all $T \in (0, T_{ε, β, R}]$ , $ε \in (0, ε_{0}]$ and $β ⩾ 0$ . We have to estimate all terms in a suitable way. First, $\frac{1}{2} \partial_{t} | {\overline{u}}_{ε} |^{2} = {\overline{u}}_{ε} \partial_{t} {\overline{u}}_{ε}$ , integration by parts in time and $\partial_{t} g_{β} = - β g_{β}$ imply $\begin{matrix} \int_{0}^{T} \int_{Ω} g_{β}^{2} \partial_{t} {\overline{u}}_{ε} {\overline{u}}_{ε} d x d t = \frac{1}{2} g_{β} {(T)}^{2} {‖ {\overline{u}}_{ε} (T) ‖}_{L^{2} (Ω)}^{2} - \frac{1}{2} {‖ {\overline{u}}_{ε} (0) ‖}_{L^{2} (Ω)}^{2} + β \int_{0}^{T} g_{β}^{2} ‖ {\overline{u}}_{ε} ‖_{L^{2} (Ω)}^{2} d t, \end{matrix}$ where $‖ {\overline{u}}_{ε} (0) ‖_{L^{2} (Ω)}^{2} ⩽ R^{2} ε^{2 M + 1}$ due to (6.4) (“well-prepared initial data”). For the other term on the left hand side in (6.8) we use integration by parts in space. This yields $\begin{array}{l} \int_{0}^{T} g_{β}^{2} \int_{Ω} {\overline{u}}_{ε} [- Δ + \frac{1}{ε^{2}} f^{″} (u_{ε}^{A})] {\overline{u}}_{ε} d x d t \\ = \int_{0}^{T} g_{β}^{2} \int_{Ω} | \nabla {\overline{u}}_{ε} |^{2} + \frac{1}{ε^{2}} f^{″} (u_{ε}^{A}) {\overline{u}}_{ε}^{2} d x d t + \int_{0}^{T} g_{β}^{2} \int_{\partial Ω} s_{ε}^{A} tr {\overline{u}}_{ε} d H^{N - 1} d t . \end{array}$ With requirement 2. (“spectral estimate”) in the theorem it follows that the first integral on the right hand side in the latter equation is bounded from below by $\begin{matrix} - C \int_{0}^{T} g_{β}^{2} ‖ {\overline{u}}_{ε} ‖_{L^{2} (Ω)}^{2} d t + ‖ g_{β} \nabla {\overline{u}}_{ε} ‖_{L^{2} (Q_{T} ∖ Γ (δ))}^{2} + c_{0} ‖ g_{β} \nabla_{τ} {\overline{u}}_{ε} ‖_{L^{2} (Q_{T} \cap Γ (δ))}^{2} . \end{matrix}$ For the remainder terms involving $r_{ε}^{A}$ and $s_{ε}^{A}$ we use (6.3) (“approximate solution”). This yields $\begin{matrix} | \int_{0}^{T} g_{β}^{2} [\int_{\partial Ω} s_{ε}^{A} tr {\overline{u}}_{ε} (t) d H^{N - 1} + \int_{Ω} r_{ε}^{A} {\overline{u}}_{ε} (t) d x] d t | ⩽ {\overline{C}}_{1} R ‖ g_{β} ‖_{L^{2} (0, T)} ε^{2 M + 1} \end{matrix}$ due to (6.5) for all $T \in (0, T_{ε, β, R}]$ and $ε \in (0, ε_{0}]$ , where we used $‖ g_{β} ‖_{L^{1} (0, T)} ⩽ \sqrt{T_{0}} ‖ g_{β} ‖_{L^{2} (0, T)}$ .

In the following we estimate the $r_{ε}$ -term in (6.8). The requirement 1. (“uniform boundedness”) in the theorem and Lemma 6.1 yield $\begin{matrix} (6.9) & sup_{ε \in (0, ε_{0}]} [‖ u_{ε} ‖_{L^{\infty} (Q_{T_{0}})} + {‖ u_{ε}^{A} ‖}_{L^{\infty} (Q_{T_{0}})}] < \infty . \end{matrix}$ Therefore we can apply the Taylor Theorem and obtain $\begin{matrix} (6.10) & | \int_{0}^{T} g_{β}^{2} \int_{Ω} r_{ε} (u_{ε}, u_{ε}^{A}) {\overline{u}}_{ε} d x d t | ⩽ \frac{C}{ε^{2}} \int_{0}^{T} g_{β}^{2} ‖ {\overline{u}}_{ε} ‖_{L^{3} (Ω)}^{3} d t . \end{matrix}$ In order to estimate the latter we use a standard Gagliardo–Nirenberg Inequality on $Ω ∖ Γ_{t} (δ)$ but on $Γ_{t} (δ)$ we apply such inequalities in tangential and normal direction. This is similar to [2, Lemma 5.3]. The idea is to get a finer estimate and account for the fact that the estimate for $\nabla_{τ} {\overline{u}}_{ε}$ in (6.5) is better than that for $\partial_{n} {\overline{u}}_{ε}$ . However, note that if N is too large, estimating the full $L^{3}$ -norm for ${\overline{u}}_{ε}$ will not work because of the requirements for the Gagliardo–Nirenberg Inequality or because we only have $L^{2}$ -estimates in (6.5) for $\nabla_{τ} {\overline{u}}_{ε}$ and $\partial_{n} {\overline{u}}_{ε}$ . Therefore we use the uniform boundedness (6.9) to lower the exponent. However, this will also decrease the resulting ε-order. Therefore we try to find the largest possible parameter. The estimates are lengthy and we decided to postpone them, see Lemma 6.7 below. The result is □
Lemma 6.7.
Under the assumptions in Theorem 6.5 it holds $\begin{matrix} | \int_{0}^{T} g_{β}^{2} \int_{Ω} r_{ε} (u_{ε}, u_{ε}^{A}) {\overline{u}}_{ε} d x d t | ⩽ C R^{2 + K (N)} ε^{2 M + 1} ε^{K (N) (M - k (N))} {‖ g_{β}^{- K (N)} ‖}_{L^{\frac{4}{4 - min {4, N}}} (0, T)} \end{matrix}$ for all $T \in (0, T_{ε, β, R}]$ and $ε \in (0, ε_{0}]$ , where $T_{ε, β, R}$ is as in ( 6.6 ), $k (N) = max {2, \frac{N}{2}}$ and $K (N) : = min {1, \frac{4}{N}} \in (0, 1]$ .
Remark 6.8.
Note that one could apply the same standard Gagliardo–Nirenberg Inequality used for $Ω ∖ Γ_{t} (δ)$ also for whole Ω, but then the estimate is weaker and the minimal order $k (N)$ for M that is required for the difference estimate to work increases.
Continuation of proof of Theorem 6.5.
It remains to estimate $\partial_{n} {\overline{u}}_{ε}$ . Therefore we use that due to Corollary 3.5 $\begin{array}{l} ε^{2} ‖ g_{β} \partial_{n} {\overline{u}}_{ε} ‖_{L^{2} (Q_{T} \cap Γ (δ))}^{2} & ⩽ C ε^{2} \int_{0}^{T} g_{β}^{2} \int_{Ω} | \nabla {\overline{u}}_{ε} |^{2} + \frac{1}{ε^{2}} f^{″} (u_{ε}^{A}) {({\overline{u}}_{ε})}^{2} d x d t \\ + C sup_{ε \in (0, ε_{0}]} {‖ f^{″} (u_{ε}^{A}) ‖}_{L^{\infty} (Q_{T_{0}})} \int_{0}^{T} g_{β}^{2} {‖ {\overline{u}}_{ε} (t) ‖}_{L^{2} (Ω)}^{2} d t \end{array}$ with a constant $C > 0$ independent of ε, T and R. The first term is absorbed with $\frac{1}{2}$ of the spectral term above if $ε \in (0, ε_{1}]$ and $ε_{1} > 0$ is small (independent of T, R). Finally, all terms are estimated and we obtain $\begin{array}{l} {‖ \frac{g_{β} {\overline{u}}_{ε}}{2} |_{T} ‖}_{L^{2} (Ω)}^{2} + {‖ \frac{g_{β}}{2} \nabla {\overline{u}}_{ε} ‖}_{L^{2} (Q_{T} ∖ Γ (δ))}^{2} + \frac{c_{0}}{2} ‖ g_{β} \nabla_{τ} {\overline{u}}_{ε} ‖_{L^{2} (Q_{T} \cap Γ (δ))}^{2} + \frac{ε^{2}}{2} ‖ g_{β} \partial_{n} {\overline{u}}_{ε} ‖_{L^{2} (Q_{T} \cap Γ (δ))}^{2} \\ ⩽ \frac{R^{2}}{2} ε^{2 M + 1} + \int_{0}^{T} (- β + {\overline{C}}_{0}) g_{β}^{2} {‖ {\overline{u}}_{ε} (t) ‖}_{L^{2} (Ω)}^{2} d t + {\overline{C}}_{1} R ε^{2 M + 1} ‖ g_{β} ‖_{L^{2} (0, T)} \\ (6.11) & + C R^{2 + K (N)} ε^{2 M + 1} ε^{K (N) (M - k (N))} {‖ g_{β}^{- K (N)} ‖}_{L^{\frac{4}{4 - min {4, N}}} (0, T)} \end{array}$ for all $T \in (0, T_{ε, β, R}]$ , $ε \in (0, ε_{1}]$ and constants ${\overline{C}}_{0}, {\overline{C}}_{1}, C > 0$ independent of $ε, T, R$ , where $k (N) = max {2, \frac{N}{2}}$ and $K (N) = min {1, \frac{4}{N}}$ .

Now we consider the cases in the theorem.
Ad 1.
If $M > k (N)$ , then we choose $β ⩾ {\overline{C}}_{0}$ large such that ${\overline{C}}_{1} R ‖ g_{β} ‖_{L^{2} (0, T_{0})} ⩽ \frac{R^{2}}{8}$ . Then (6.11) is estimated by $\frac{3}{4} R^{2} ε^{2 M + 1}$ for all $T \in (0, T_{ε, β, R}]$ and $ε \in (0, ε_{1}]$ , if $ε_{1} > 0$ is small. By contradiction and continuity this shows $T_{ε, β, R} = T_{0}$ for all $ε \in (0, ε_{1}]$ . □
Ad 2.
Let $k (N) \in N$ , $M = k (N)$ and let (6.3) hold for some $\tilde{M} > M$ instead of M. Then the term in (6.11) where R enters linearly is improved by a factor $ε^{\tilde{M} - M}$ . Let $β ⩾ {\overline{C}}_{0}$ be fixed. Now we can first choose $R > 0$ small such that the $R^{2 + K (N)}$ -term in (6.11) is bounded by $\frac{1}{8} R^{2} ε^{2 M + 1}$ . Then $ε_{1} > 0$ can be taken small such that (6.11) is estimated by $\frac{3}{4} R^{2} ε^{2 M + 1}$ for all $T \in (0, T_{ε, β, R}]$ and $ε \in (0, ε_{1}]$ . By contradiction we get $T_{ε, β, R} = T_{0}$ for all $ε \in (0, ε_{1}]$ . □
Ad 3.
Finally, let $N \in {2, 3}$ , $M = 2$ and $β = 0$ . Then (6.11) is dominated by $\begin{matrix} [\frac{R^{2}}{2} + C R^{2} T + C R T^{\frac{1}{2}} + C R^{3} T^{\frac{4 - N}{4}}] ε^{2 M + 1} . \end{matrix}$ Due to $\frac{4 - N}{4} > 0$ there are $ε_{1}, T_{1} > 0$ such that this is bounded by $\frac{3}{4} R^{2} ε^{2 M + 1}$ for every $T \in (0, min (T_{ε, β, R}, T_{1})]$ and $ε \in (0, ε_{1}]$ . Therefore $T_{ε, 0, R} ⩾ T_{1}$ for all $ε \in (0, ε_{1}]$ . □

The proof of Theorem 6.5 is completed. □ Proof of Lemma 6.7.
First, let us estimate (6.10) for $Ω ∖ Γ_{t} (δ)$ instead of Ω. The Gagliardo–Nirenberg Inequality in Lemma 6.3 and Remark 6.4 imply for $2 ⩽ N ⩽ 6$ $\begin{matrix} {‖ {\overline{u}}_{ε} (t) ‖}_{L^{3} (Ω ∖ Γ_{t} (δ))}^{3} ⩽ C {‖ {\overline{u}}_{ε} (t) ‖}_{L^{2} (Ω ∖ Γ_{t} (δ))}^{3 (1 - \frac{N}{6})} {‖ {\overline{u}}_{ε} (t) ‖}_{H^{1} (Ω ∖ Γ_{t} (δ))}^{3 \frac{N}{6}}, \end{matrix}$ where the constant C is independent of $t \in [0, T_{0}]$ because of Remark 6.4 and since $Ω ∖ Γ_{t} (δ)$ has a Lipschitz-boundary uniformly in $t \in [0, T_{0}]$ , cf. [66, Remark 6.29]. For $3 \frac{N}{6} > 2$ , i.e. $N = 5, 6$ , the right hand side can not be controlled with (6.5). But for $N \in {2, 3, 4}$ we can use this estimate and obtain that (6.10) for $Ω ∖ Γ_{t} (δ)$ instead of Ω is estimated by $\begin{matrix} (6.12) & \begin{matrix} \frac{C}{ε^{2}} \int_{0}^{T} g_{β}^{- 1} [g_{β}^{3} {‖ {\overline{u}}_{ε} (t) ‖}_{L^{2} (Ω ∖ Γ_{t} (δ))}^{3} + g_{β}^{3 - \frac{N}{2}} ‖ {\overline{u}}_{ε} ‖_{L^{2} (Ω ∖ Γ_{t} (δ))}^{3 - \frac{N}{2}} g_{β}^{\frac{N}{2}} {‖ \nabla {\overline{u}}_{ε} (t) ‖}_{L^{2} (Ω ∖ Γ_{t} (δ))}^{\frac{N}{2}}] d t \\ ⩽ C R^{3} ε^{- 2 + 3 (M + \frac{1}{2})} [{‖ g_{β}^{- 1} ‖}_{L^{1} (0, T)} + {‖ g_{β}^{- 1} ‖}_{L^{\frac{4}{4 - N}} (0, T)}] \end{matrix} \end{matrix}$ for all $T \in (0, T_{ε, β, R}]$ and $ε \in (0, ε_{0}]$ , where we used (6.5) and the Hölder Inequality with exponents ∞, $\frac{4}{N}$ and $\frac{4}{4 - N}$ for the second term. Now let $N ⩾ 5$ . Then we consider $r (N) \in (2, 3]$ (to be determined later) and estimate with the uniform boundedness from (6.9) $\begin{matrix} {‖ {\overline{u}}_{ε} (t) ‖}_{L^{3} (Ω ∖ Γ_{t} (δ))}^{3} ⩽ C {‖ {\overline{u}}_{ε} (t) ‖}_{L^{r (N)} (Ω ∖ Γ_{t} (δ))}^{r (N)} . \end{matrix}$ By Lemma 6.3 the Gagliardo–Nirenberg Inequality is applicable for all $θ = θ (N) \in [0, 1]$ with $\begin{matrix} θ (\frac{1}{2} - \frac{1}{N}) + \frac{1 - θ}{2} = \frac{1}{r (N)} \Leftrightarrow θ = N (\frac{1}{2} - \frac{1}{r (N)}) . \end{matrix}$ The condition $θ \in [0, 1]$ restricts $r (N)$ to be in $(2, 2 + \frac{4}{N - 2}]$ . Then Lemma 6.3 and Remark 6.4 yield $\begin{matrix} {‖ {\overline{u}}_{ε} (t) ‖}_{L^{r (N)} (Ω ∖ Γ_{t} (δ))}^{r (N)} ⩽ C {‖ {\overline{u}}_{ε} (t) ‖}_{L^{2} (Ω ∖ Γ_{t} (δ))}^{r (N) (1 - θ)} {‖ {\overline{u}}_{ε} (t) ‖}_{H^{1} (Ω ∖ Γ_{t} (δ))}^{r (N) θ} . \end{matrix}$ Because we have to use (6.5) to control the right hand side, we require $r (N) θ ⩽ 2$ . This is equivalent to $\begin{matrix} N (\frac{r (N)}{2} - 1) ⩽ 2 \Leftrightarrow r (N) ⩽ 2 + \frac{4}{N} . \end{matrix}$ Since $2 + \frac{4}{N} ⩽ min {3, 2 + \frac{4}{N - 2}}$ for $N ⩾ 5$ , we can take $r (N) : = 2 + \frac{4}{N}$ for $N ⩾ 5$ . Therefore $r (N) θ = 2$ and $r (N) (1 - θ) = r (N) - r (N) θ = \frac{4}{N}$ . Hence for $N ⩾ 5$ we obtain the estimate $\begin{matrix} {‖ {\overline{u}}_{ε} (t) ‖}_{L^{3} (Ω ∖ Γ_{t} (δ))}^{3} ⩽ C {‖ {\overline{u}}_{ε} (t) ‖}_{L^{2} (Ω ∖ Γ_{t} (δ))}^{\frac{4}{N}} {‖ {\overline{u}}_{ε} (t) ‖}_{H^{1} (Ω ∖ Γ_{t} (δ))}^{2} . \end{matrix}$ Note that for $N = 4$ the calculation also works but yields the same as before. Hence using (6.5) we obtain for $N ⩾ 5$ that (6.10) with Ω replaced by $Ω ∖ Γ_{t} (δ)$ is estimated via $\begin{array}{l} \frac{C}{ε^{2}} \int_{0}^{T} g_{β}^{- \frac{4}{N}} [g_{β}^{2 + \frac{4}{N}} {‖ {\overline{u}}_{ε} (t) ‖}_{L^{2} (Ω ∖ Γ_{t} (δ))}^{2 + \frac{4}{N}} + g_{β}^{\frac{4}{N}} {‖ {\overline{u}}_{ε} (t) ‖}_{L^{2} (Ω ∖ Γ_{t} (δ))}^{\frac{4}{N}} g_{β}^{2} {‖ \nabla {\overline{u}}_{ε} (t) ‖}_{L^{2} (Ω ∖ Γ_{t} (δ))}^{2}] d t \\ (6.13) & ⩽ C R^{2 + \frac{4}{N}} ε^{- 2 + (2 + \frac{4}{N}) (M + \frac{1}{2})} [{‖ g_{β}^{- \frac{4}{N}} ‖}_{L^{1} (0, T)} + {‖ g_{β}^{- \frac{4}{N}} ‖}_{L^{\infty} (0, T)}] \end{array}$ for all $T \in (0, T_{ε, β, R}]$ and $ε \in (0, ε_{0}]$ .

Next we estimate (6.10) for $Γ_{t} (δ)$ instead of Ω. First we transform the integral to $(- δ, δ) \times Σ$ via $\overline{X}$ . This yields $\begin{matrix} \int_{0}^{T} g_{β}^{2} ‖ {\overline{u}}_{ε} ‖_{L^{3} (Ω)}^{3} d t = \int_{0}^{T} g_{β}^{2} \int_{- δ}^{δ} \int_{Σ} | {\overline{u}}_{ε} |_{\overline{X} (r, s, t)} |^{3} J_{t} (r, s) d H^{N - 1} (s) d r d t . \end{matrix}$ Here $0 < c ⩽ J_{t} ⩽ C$ with $c, C > 0$ independent of t by Remark 3.3, 3. Because of Lemma 2.3 and Remark 6.4 we can use the Gagliardo–Nirenberg Inequality for Σ with $N - 1$ instead of N (and full $W^{1, p}$ -norm on the right hand side). First we consider $N \in {2, 3, 4}$ since this was a special case for the estimate on $Ω ∖ Γ_{t} (δ)$ , too. Then $\begin{matrix} \int_{0}^{T} g_{β}^{2} \int_{- δ}^{δ} ‖ {\overline{u}}_{ε} |_{\overline{X} (r, \cdot, t)} ‖_{L^{3} (Σ)}^{3} d r d t ⩽ \int_{0}^{T} C g_{β}^{2} \int_{- δ}^{δ} ‖ {\overline{u}}_{ε} |_{\overline{X} (r, \cdot, t)} ‖_{L^{2} (Σ)}^{\frac{7 - N}{2}} ‖ {\overline{u}}_{ε} |_{\overline{X} (r, \cdot, t)} ‖_{H^{1} (Σ)}^{\frac{N - 1}{2}} d r d t . \end{matrix}$ Due to properties of Sobolev-spaces on product sets, cf. [66, Lemma 2.17], we can use the Hölder Inequality with exponents $\frac{4}{5 - N}$ and $\frac{4}{N - 1}$ . Therefore $\begin{matrix} \int_{- δ}^{δ} ‖ {\overline{u}}_{ε} |_{\overline{X} (r, \cdot, t)} ‖_{L^{3} (Σ)}^{3} d r ⩽ C ‖ {\overline{u}}_{ε} |_{\overline{X} (\cdot, t)} ‖_{L^{2 \frac{7 - N}{5 - N}} (- δ, δ, L^{2} (Σ))}^{\frac{7 - N}{2}} ‖ {\overline{u}}_{ε} |_{\overline{X} (\cdot, t)} ‖_{L^{2} (- δ, δ, H^{1} (Σ))}^{\frac{N - 1}{2}} . \end{matrix}$ Here $2 \frac{7 - N}{5 - N} \in (2, \infty)$ . Hence for the first term we can use the Gagliardo–Nirenberg Inequality for $(- δ, δ)$ . The condition on the intermediate parameter $θ = θ (N) \in [0, 1]$ is $\begin{matrix} θ (\frac{1}{2} - 1) + \frac{1 - θ}{2} = \frac{5 - N}{2 (7 - N)} \Leftrightarrow θ = \frac{1}{7 - N} . \end{matrix}$ Hence we obtain $\begin{matrix} ‖ {\overline{u}}_{ε} |_{\overline{X} (\cdot, t)} ‖_{L^{2 \frac{7 - N}{5 - N}} (- δ, δ, L^{2} (Σ))}^{\frac{7 - N}{2}} ⩽ C ‖ {\overline{u}}_{ε} |_{\overline{X} (\cdot, t)} ‖_{L^{2} (- δ, δ, L^{2} (Σ))}^{\frac{6 - N}{2}} ‖ {\overline{u}}_{ε} |_{\overline{X} (\cdot, t)} ‖_{H^{1} (- δ, δ, L^{2} (Σ))}^{\frac{1}{2}} . \end{matrix}$ Hence (6.10) for $Γ_{t} (δ)$ instead of Ω and $N \in {2, 3, 4}$ is estimated by $\begin{matrix} (6.14) & \frac{C}{ε^{2}} \int_{0}^{T} g_{β}^{2} ‖ {\overline{u}}_{ε} |_{\overline{X} (\cdot, t)} ‖_{L^{2} (- δ, δ, L^{2} (Σ))}^{\frac{6 - N}{2}} ‖ {\overline{u}}_{ε} |_{\overline{X} (\cdot, t)} ‖_{H^{1} (- δ, δ, L^{2} (Σ))}^{\frac{1}{2}} ‖ {\overline{u}}_{ε} |_{\overline{X} (\cdot, t)} ‖_{L^{2} (- δ, δ, H^{1} (Σ))}^{\frac{N - 1}{2}} d t . \end{matrix}$ Moreover, because of Lemma 2.3, properties of Sobolev spaces on product sets in [66, Lemma 2.17] and Corollary 3.5, 1. it holds $\begin{array}{l} ‖ {\overline{u}}_{ε} |_{\overline{X} (\cdot, t)} ‖_{H^{1} (- δ, δ, L^{2} (Σ))} ⩽ C (‖ {\overline{u}}_{ε} |_{\overline{X} (\cdot, t)} ‖_{L^{2} ((- δ, δ) \times Σ)} + ‖ \partial_{n} {\overline{u}}_{ε} |_{\overline{X} (\cdot, t)} ‖_{L^{2} ((- δ, δ) \times Σ)}), \\ ‖ {\overline{u}}_{ε} |_{\overline{X} (\cdot, t)} ‖_{L^{2} (- δ, δ, H^{1} (Σ))} ⩽ C (‖ {\overline{u}}_{ε} |_{\overline{X} (\cdot, t)} ‖_{L^{2} ((- δ, δ) \times Σ)} + ‖ \nabla_{τ} {\overline{u}}_{ε} |_{\overline{X} (\cdot, t)} ‖_{L^{2} ((- δ, δ) \times Σ)}) . \end{array}$ For the product term in (6.14) involving both $\partial_{n} {\overline{u}}_{ε}$ and $\nabla_{τ} {\overline{u}}_{ε}$ as factors we apply the Hölder Inequality with exponents $\frac{4}{4 - N}$ , 4 and $\frac{4}{N - 1}$ . For the term with $\partial_{n} {\overline{u}}_{ε}$ we use the exponents $\frac{4}{3}$ , 4 and for the one with $\nabla_{τ}$ we use $\frac{4}{5 - N}$ , $\frac{4}{N - 1}$ . Altogether (6.10) for $Γ_{t} (δ)$ instead of Ω and $N \in {2, 3, 4}$ is controlled by $\begin{matrix} (6.15) & C R^{3} ε^{3 M - 1} [{‖ g_{β}^{- 1} ‖}_{L^{1} (0, T)} + {‖ g_{β}^{- 1} ‖}_{L^{\frac{4}{5 - N}} (0, T)} + {‖ g_{β}^{- 1} ‖}_{L^{\frac{4}{3}} (0, T)} + {‖ g_{β}^{- 1} ‖}_{L^{\frac{4}{4 - N}} (0, T)}] \end{matrix}$ for all $T \in (0, T_{ε, β, R}]$ and $ε \in (0, ε_{0}]$ , where we used (6.5) and $ε ⩽ ε_{0}$ for the terms that possess a higher ε-order. Now let $N ⩾ 5$ . Then with (6.9) we estimate $\begin{matrix} {‖ {\overline{u}}_{ε} (t) ‖}_{L^{3} (Γ_{t} (δ))}^{3} ⩽ {‖ {\overline{u}}_{ε} (t) ‖}_{L^{\tilde{r} (N)} (Γ_{t} (δ))}^{\tilde{r} (N)} \end{matrix}$ for some $\tilde{r} (N) \in (2, 3]$ . We seek the maximal $\tilde{r} (N)$ such that similar calculations as above work. It will turn out that $\tilde{r} (N) = 2 + \frac{4}{N}$ is optimal. Note that the latter also was the best exponent for the estimate on $Ω ∖ Γ_{t} (δ)$ above in the case $N ⩾ 5$ . But let us carry out the calculations with a general $\tilde{r} (N)$ . The Gagliardo–Nirenberg Inequality on Σ is applicable for $θ = θ (N) \in [0, 1]$ with $\begin{matrix} θ (\frac{1}{2} - \frac{1}{N - 1}) + \frac{1 - θ}{2} = \frac{1}{\tilde{r} (N)} \Leftrightarrow θ = (N - 1) (\frac{1}{2} - \frac{1}{\tilde{r} (N)}) . \end{matrix}$ This restricts $\tilde{r} (N)$ to be in $(2, 2 + \frac{4}{N - 3}]$ . Therefore for such $\tilde{r} (N)$ we obtain the estimate $\begin{matrix} \int_{0}^{T} g_{β}^{2} \int_{- δ}^{δ} ‖ {\overline{u}}_{ε} |_{\overline{X} (r, \cdot, t)} ‖_{L^{\tilde{r} (N)} (Σ)}^{\tilde{r} (N)} d r d t ⩽ \int_{0}^{T} C g_{β}^{2} \int_{- δ}^{δ} ‖ {\overline{u}}_{ε} |_{\overline{X} (r, \cdot, t)} ‖_{L^{2} (Σ)}^{q (N)} ‖ {\overline{u}}_{ε} |_{\overline{X} (r, \cdot, t)} ‖_{H^{1} (Σ)}^{p (N)} d r d t, \end{matrix}$ where $p (N) : = θ \tilde{r} (N) = (N - 1) (\frac{\tilde{r} (N)}{2} - 1)$ and $q (N) : = (1 - θ) \tilde{r} (N) = \tilde{r} (N) - p (N)$ . The next step is to use the Hölder Inequality on $(- δ, δ)$ . To this end we need $p (N) ⩽ 2$ . This is equivalent to $\tilde{r} (N) ⩽ 2 + \frac{4}{N - 1}$ . Hence for these $\tilde{r} (N)$ we can use the Hölder Inequality with exponents $\frac{2}{2 - p (N)}$ , $\frac{2}{p (N)}$ and obtain $\begin{matrix} \int_{- δ}^{δ} ‖ {\overline{u}}_{ε} |_{\overline{X} (r, \cdot, t)} ‖_{L^{\tilde{r} (N)} (Σ)}^{\tilde{r} (N)} d r ⩽ C ‖ {\overline{u}}_{ε} |_{\overline{X} (\cdot, t)} ‖_{L^{y (N)} (- δ, δ, L^{2} (Σ))}^{q (N)} ‖ {\overline{u}}_{ε} |_{\overline{X} (\cdot, t)} ‖_{L^{2} (- δ, δ, H^{1} (Σ))}^{p (N)}, \end{matrix}$ where we have set $y (N) : = \frac{2 q (N)}{2 - p (N)}$ . Note that $y (N) = \frac{2 \tilde{r} (N) - 2 p (N)}{2 - p (N)} > 2$ . Therefore we can use the Gagliardo–Nirenberg Inequality on $(- δ, δ)$ . The condition for $θ = θ (N) \in [0, 1]$ is $\begin{matrix} θ (\frac{1}{2} - 1) + \frac{θ}{2} = \frac{1}{y (N)} \Leftrightarrow θ = \frac{1}{2} - \frac{1}{y (N)} = \frac{q (N) + p (N) - 2}{2 q (N)} = \frac{\tilde{r} (N) - 2}{2 q (N)} . \end{matrix}$ Hence we obtain $\begin{matrix} ‖ {\overline{u}}_{ε} |_{\overline{X} (\cdot, t)} ‖_{L^{y (N)} (- δ, δ, L^{2} (Σ))}^{q (N)} ⩽ C ‖ {\overline{u}}_{ε} |_{\overline{X} (\cdot, t)} ‖_{L^{2} (- δ, δ, L^{2} (Σ))}^{q (N) + 1 - \frac{\tilde{r} (N)}{2}} ‖ {\overline{u}}_{ε} |_{\overline{X} (\cdot, t)} ‖_{H^{1} (- δ, δ, L^{2} (Σ))}^{\frac{\tilde{r} (N)}{2} - 1} . \end{matrix}$ Therefore (6.10) for $Γ_{t} (δ)$ instead of Ω and $N ⩾ 5$ is estimated by $\begin{matrix} (6.16) & \frac{C}{ε^{2}} \int_{0}^{T} g_{β}^{2} ‖ {\overline{u}}_{ε} |_{\overline{X} (\cdot, t)} ‖_{L^{2} (- δ, δ, L^{2} (Σ))}^{1 + q (N) - \frac{\tilde{r} (N)}{2}} ‖ {\overline{u}}_{ε} |_{\overline{X} (\cdot, t)} ‖_{H^{1} (- δ, δ, L^{2} (Σ))}^{\frac{\tilde{r} (N)}{2} - 1} ‖ {\overline{u}}_{ε} |_{\overline{X} (\cdot, t)} ‖_{L^{2} (- δ, δ, H^{1} (Σ))}^{p (N)} d t . \end{matrix}$ We can estimate the terms using $\partial_{n}$ and $\nabla_{τ}$ , see below (6.14). The last step is to use the Hölder Inequality in time. For the product of the $\partial_{n} {\overline{u}}_{ε}$ -term and the $\nabla_{τ} {\overline{u}}_{ε}$ -term we want to use the Hölder Inequality with exponents $2 \frac{2}{\tilde{r} (N) - 2}$ , $\frac{2}{p (N)}$ . This gives the following condition for $\tilde{r} (N)$ : $\begin{matrix} \frac{p (N)}{2} + \frac{\tilde{r} (N) - 2}{2} ⩽ 2 \Leftrightarrow \tilde{r} (N) ⩽ 2 + \frac{4}{N} . \end{matrix}$ For $\tilde{r} (N) = 2 + \frac{4}{N}$ all the calculations work and $p (N) = 2 - \frac{2}{N} = 2 \frac{N - 1}{N}$ , $q (N) = \frac{6}{N}$ as well as $1 + q (N) - \frac{\tilde{r} (N)}{2} = \frac{4}{N}$ and $\frac{\tilde{r} (N)}{2} - 1 = \frac{2}{N}$ . Altogether (6.10) for $Γ_{t} (δ)$ instead of Ω and $N ⩾ 5$ is controlled by $C R^{2 + \frac{4}{N}}$ times $\begin{matrix} (6.17) & ε^{- 2 + (2 + \frac{4}{N}) M + 1} [{‖ g_{β}^{- \frac{4}{N}} ‖}_{L^{1} (0, T)} + {‖ g_{β}^{- \frac{4}{N}} ‖}_{L^{N} (0, T)} + {‖ g_{β}^{- \frac{4}{N}} ‖}_{L^{\frac{N}{N - 1}} (0, T)} + {‖ g_{β}^{- \frac{4}{N}} ‖}_{L^{\infty} (0, T)}] \end{matrix}$ for all $T \in (0, T_{ε, β, R}]$ and $ε \in (0, ε_{0}]$ , where we used (6.5) and $ε ⩽ ε_{0}$ .

Finally, we collect the above estimates. To reduce the number of $g_{β}$ -terms we apply the embedding $L^{p} (0, t) ↪ L^{q} (0, t)$ for all $1 ⩽ q ⩽ p ⩽ \infty$ and $0 < t ⩽ T_{0}$ with embedding constant independent of t. Therefore if $N \in {2, 3, 4}$ , then by (6.12), (6.15), and if $N ⩾ 5$ , then by (6.13), (6.17), we get $\begin{matrix} | \int_{0}^{T} g_{β}^{2} \int_{Ω} r_{ε} (u_{ε}, u_{ε}^{A}) {\overline{u}}_{ε} d x d t | ⩽ \{\begin{matrix} C R^{3} ε^{3 M - 1} ‖ g_{β}^{- 1} ‖_{L^{\frac{4}{4 - N}} (0, T)} & for N \in {2, 3, 4}, \\ C R^{2 + \frac{4}{N}} ε^{(2 + \frac{4}{N}) M - 1} ‖ g_{β}^{- \frac{4}{N}} ‖_{L^{\infty} (0, T)} & for N ⩾ 4 . \end{matrix} \end{matrix}$ This shows Lemma 6.7. □

6.2.2. Proof of Theorem 1.4

Let $N ⩾ 2$ , Ω, $Q_{T}$ and $\partial Q_{T}$ be as in Remark 1.3, 1. Moreover, let $Γ = {(Γ_{t})}_{t \in [0, T_{0}]}$ for some $T_{0} > 0$ be a smooth solution to (MCF) with 90°-contact angle condition parametrized as in Section 3.1 and let $δ > 0$ be such that Theorem 3.2 holds for $2 δ$ instead of δ. We use the notation from Section 3.1 and Section 3.2. Let $M \in N$ with $M ⩾ 2$ and denote with ${(u_{ε}^{A})}_{ε > 0}$ the approximate solution on $\overline{Q_{T_{0}}}$ defined in Section 5.1.3 (which we obtained from asymptotic expansions in Section 5.1) and let $ε_{0} > 0$ be such that Lemma 5.10 (“remainder estimate”) holds for $ε \in (0, ε_{0}]$ . The property ${lim}_{ε \to 0} u_{ε}^{A} = \pm 1$ uniformly on compact subsets of $Q_{T_{0}}^{\pm}$ follows from the construction in Section 5.1.

Note that $g_{β} (t) : = e^{- β t}$ for fixed β is trapped between uniform positive constants for all $t \in [0, T_{0}]$ . Therefore Theorem 1.4 follows immediately from Theorem 6.5 if we show the conditions 1.–4. in Theorem 6.5. The requirement 1. (“uniform boundedness”) is fulfilled due to Lemma 5.10 for $u_{ε}^{A}$ and for $u_{0, ε}$ this is an assumption in Theorem 1.4, Condition 2. (“spectral estimate”) is valid because of [67, Theorem 1.5]. Requirement 4. (“well prepared initial data”) is a condition on $u_{0, ε}$ and assumed in Theorem 1.4. It remains to prove 3. (“approximate solution”).

First we estimate the boundary term in (6.3). Lemma 5.10 yields $s_{ε}^{A} = 0$ on $\partial Ω ∖ Γ_{t} (2 δ)$ and $| s_{ε}^{A} | ⩽ C ε^{M} e^{- c | ρ_{ε} |}$ , where $ρ_{ε}$ is defined in (5.1). Therefore $\begin{matrix} | \int_{\partial Ω} s_{ε}^{A} tr {\overline{u}}_{ε} (t) d H^{N - 1} | ⩽ {‖ s_{ε}^{A} ‖}_{L^{2} (\partial Ω \cap Γ_{t} (2 δ))} {‖ tr {\overline{u}}_{ε} (t) ‖}_{L^{2} (\partial Ω \cap Γ_{t} (2 δ))} . \end{matrix}$ Here by the substitution rule it holds $\begin{matrix} {‖ s_{ε}^{A} ‖}_{L^{2} (\partial Ω \cap Γ_{t} (2 δ))}^{2} = \int_{\partial Σ} \int_{- 2 δ}^{2 δ} {| s_{ε}^{A} |}^{2} |_{\overline{X} (r, Y (σ, 0), t)} | det d_{(r, σ)} [X (\cdot, Y (\cdot, 0), t)] | d r d H^{N - 2} (σ) . \end{matrix}$ With a scaling argument this is estimated by $C ε^{2 M + 1}$ . If required the reader may compare with [67, Lemma 5.4]. Moreover, one can prove $\begin{matrix} {‖ tr {\overline{u}}_{ε} (t) ‖}_{L^{2} (\partial Ω \cap Γ_{t} (2 δ))} ⩽ C ({‖ {\overline{u}}_{ε} (t) ‖}_{L^{2} (Γ_{t} (2 δ))} + {‖ \nabla_{τ} {\overline{u}}_{ε} (t) ‖}_{L^{2} (Γ_{t} (2 δ))}) . \end{matrix}$ This can be shown with a similar idea as in Evans [35, 5.10, problem 7]. See also [67, Lemma 5.22]. Here one uses $\vec{w}$ in the proof of the latter with $w_{1} : = 0$ there and then Corollary 3.5 to estimate $| \nabla_{Σ} ({\overline{u}}_{ε} |_{\overline{X}}) | ⩽ C | \nabla_{τ} {\overline{u}}_{ε} |_{\overline{X}} |$ . Moreover, $| \nabla_{τ} {\overline{u}}_{ε} | ⩽ C | \nabla {\overline{u}}_{ε} |$ by Corollary 3.5 and the estimate for the $s_{ε}^{A}$ -term in (6.3) follows.

Finally, we estimate the $r_{ε}^{A}$ -term in (6.3). Lemma 5.10 yields $r_{ε}^{A} = 0$ in $Ω ∖ Γ_{t} (2 δ)$ and $\begin{array}{l} | r_{ε}^{A} | ⩽ C (ε^{M} e^{- c | ρ_{ε} |} + ε^{M + 1}) in Γ (2 δ, μ_{1}), \\ | r_{ε}^{A} | ⩽ C (ε^{M - 1} e^{- c (| ρ_{ε} | + H_{ε})} + ε^{M} e^{- c | ρ_{ε} |} + ε^{M + 1}) in Γ^{C} (2 δ, 2 μ_{1}) . \end{array}$ The substitution rule implies $\begin{matrix} | \int_{Ω} r_{ε}^{A} {\overline{u}}_{ε} (t) d x | ⩽ \int_{Γ_{t} (2 δ)} | r_{ε}^{A} {\overline{u}}_{ε} (t) | d x = \int_{Σ} \int_{- 2 δ}^{2 δ} {| r_{ε}^{A} {\overline{u}}_{ε} |}_{\overline{X} (r, s, t)} | J_{t} (r, s) d r d H^{N - 1} (s), \end{matrix}$ where $J_{t}$ is uniformly bounded in $t \in [0, T_{0}]$ by Remark 3.3, 3. We split the integral over Σ in integrals over $Σ ∖ Y (\partial Σ \times [0, 2 μ_{1}])$ and $Y (\partial Σ \times [0, 2 μ_{1}])$ . For both we use the Hölder Inequality with exponents 2, 2 for the inner integral. With a scaling argument and Hölder’s inequality the integral over $Σ ∖ Y (\partial Σ \times [0, 2 μ_{1}])$ is estimated by $\begin{matrix} C ε^{M + \frac{1}{2}} \int_{Σ ∖ Y (\partial Σ \times [0, 2 μ_{1}])} ‖ {\overline{u}}_{ε} |_{\overline{X} (\cdot, s, t)} ‖_{L^{2} (- 2 δ, 2 δ)} d H^{N - 1} (s) ⩽ C ε^{M + \frac{1}{2}} {‖ {\overline{u}}_{ε} (t) ‖}_{L^{2} (Γ_{t} (2 δ, 2 μ_{1}))} . \end{matrix}$ Moreover, by the substitution rule the integral over $Y (\partial Σ \times [0, 2 μ_{1}])$ is controlled via $\begin{matrix} C ε^{M - \frac{1}{2}} \int_{\partial Σ} \int_{0}^{2 μ_{1}} ‖ {\overline{u}}_{ε} |_{\overline{X} (\cdot, Y (σ, b), t)} ‖_{L^{2} (- 2 δ, 2 δ)} (e^{- c \frac{b}{ε}} + ε) d b d H^{N - 2} (σ) . \end{matrix}$ For the ${\overline{u}}_{ε}$ -term we use $H^{1} (0, 2 μ_{1}, L^{2} (- 2 δ, 2 δ)) ↪ L^{\infty} (0, 2 μ_{1}, L^{2} (- 2 δ, 2 δ))$ . Moreover, a scaling argument yields $\int_{0}^{2 μ_{1}} e^{- c b / ε} d b ⩽ C ε$ . Hence the above term is estimated by $\begin{matrix} C ε^{M + \frac{1}{2}} \int_{\partial Σ} ‖ {\overline{u}}_{ε} |_{\overline{X} (\cdot, Y (σ, \cdot), t)} ‖_{L^{2} (- 2 δ, 2 δ, H^{1} (0, 2 μ_{1}))} d H^{N - 2} (σ) . \end{matrix}$ Note that due to well-known properties of Sobolev spaces on product sets the expression $‖ {\overline{u}}_{ε} |_{\overline{X} (\cdot, Y (σ, \cdot), t)} ‖_{L^{2} (- 2 δ, 2 δ, H^{1} (0, 2 μ_{1}))}$ equals $\begin{matrix} ‖ {\overline{u}}_{ε} |_{\overline{X} (\cdot, Y (σ, \cdot), t)} ‖_{L^{2} ((- 2 δ, 2 δ) \times (0, 2 μ_{1}))} + ‖ \partial_{b} {\overline{u}}_{ε} |_{\overline{X} (\cdot, Y (σ, \cdot), t)} ‖_{L^{2} ((- 2 δ, 2 δ) \times (0, 2 μ_{1}))} \end{matrix}$ and Corollary 3.5 yields $\begin{matrix} ‖ \partial_{b} {\overline{u}}_{ε} |_{\overline{X} (\cdot, Y (σ, \cdot), t)} ‖_{L^{2} ((- 2 δ, 2 δ) \times (0, 2 μ_{1}))} ⩽ C ‖ \nabla_{τ} {\overline{u}}_{ε} |_{\overline{X} (\cdot, Y (σ, \cdot), t)} ‖_{L^{2} ((- 2 δ, 2 δ) \times (0, 2 μ_{1}))} . \end{matrix}$ Finally, by the substitution rule and Hölder’s inequality we obtain $\begin{matrix} | \int_{Ω} r_{ε}^{A} {\overline{u}}_{ε} (t) d x | ⩽ C ε^{M + \frac{1}{2}} ({‖ {\overline{u}}_{ε} (t) ‖}_{L^{2} (Γ_{t} (2 δ))} + {‖ \nabla_{τ} {\overline{u}}_{ε} (t) ‖}_{L^{2} (Γ_{t} (2 δ))}) . \end{matrix}$ The estimate $| \nabla_{τ} {\overline{u}}_{ε} | ⩽ C | \nabla {\overline{u}}_{ε} |$ due to Corollary 3.5 yields (6.3). Therefore Theorem 1.4 follows from the difference estimates in Theorem 6.5.

6.3. Difference estimate and proof of the convergence theorem for (vAC) in ND

We show in Section 6.3.1 the difference estimate for exact and suitable approximate solutions for the vector-valued Allen–Cahn equation (vAC1)–(vAC3). Then in Section 6.3.2 we prove the Theorem 1.6 about convergence by checking the requirements for the difference estimate applied to the approximate solution from Section 5.2.3. All computations are analogous to the scalar case in the last Section 6.2.

6.3.1. Difference estimate

Theorem 6.9 (Difference Estimate for (vAC)).

Moreover, let ${\overset{ˇ}{ε}}_{0} > 0$ , ${\vec{u}}_{ε}^{A} \in C^{2} {(\overline{Q_{T_{0}}})}^{m}$ , ${\vec{u}}_{0, ε} \in C^{2} {(\overline{Ω})}^{m}$ with $\partial_{N_{\partial Ω}} {\vec{u}}_{0, ε} = 0$ on $\partial Ω$ and let ${\vec{u}}_{ε} \in C^{2} {(\overline{Q_{T_{0}}})}^{m}$ be exact solutions to ( vAC1 )–( vAC3 ) with ${\vec{u}}_{0, ε}$ in ( vAC3 ) for $ε \in (0, {\overset{ˇ}{ε}}_{0}]$ .

For some $R > 0$ and $M \in N, M ⩾ k (N) : = max {2, \frac{N}{2}}$ we impose the following conditions:

Uniform Boundedness: ${sup}_{ε \in (0, {\overset{ˇ}{ε}}_{0}]} ‖ {\vec{u}}_{ε}^{A} ‖_{L^{\infty} {(Q_{T_{0}})}^{m}} + ‖ {\vec{u}}_{0, ε} ‖_{L^{\infty} {(Ω)}^{m}} < \infty$ .

Spectral Estimate: There are ${\overset{ˇ}{c}}_{0}, \overset{ˇ}{C} > 0$ such that $\begin{array}{l} \int_{Ω} | \nabla \vec{ψ} |^{2} + \frac{1}{ε^{2}} {(\vec{ψ}, D^{2} W ({\vec{u}}_{ε}^{A} (\cdot, t)) \vec{ψ})}_{R^{m}} d x \\ ⩾ - \overset{ˇ}{C} ‖ \vec{ψ} ‖_{L^{2} {(Ω)}^{m}}^{2} + ‖ \nabla \vec{ψ} ‖_{L^{2} {(Ω ∖ Γ_{t} (δ))}^{N \times m}}^{2} + {\overset{ˇ}{c}}_{0} ‖ \nabla_{τ} \vec{ψ} ‖_{L^{2} {(Γ_{t} (δ))}^{N \times m}}^{2} \end{array}$ for all $\vec{ψ} \in H^{1} {(Ω)}^{m}$ and $ε \in (0, {\overset{ˇ}{ε}}_{0}], t \in [0, T_{0}]$ .

Approximate Solution: For the remainders $\begin{matrix} {\vec{r}}_{ε}^{A} : = \partial_{t} {\vec{u}}_{ε}^{A} - Δ {\vec{u}}_{ε}^{A} + \frac{1}{ε^{2}} \nabla W ({\vec{u}}_{ε}^{A}) and {\vec{s}}_{ε}^{A} : = \partial_{N_{\partial Ω}} {\vec{u}}_{ε}^{A} \end{matrix}$ in ( vAC1 )–( vAC2 ) for ${\vec{u}}_{ε}^{A}$ and the difference ${\underline{u}}_{ε} : = {\vec{u}}_{ε} - {\vec{u}}_{ε}^{A}$ it holds $\begin{matrix} (6.18) & \begin{matrix} | \int_{\partial Ω} {\vec{s}}_{ε}^{A} \cdot tr {\underline{u}}_{ε} (t) d H^{N - 1} + \int_{Ω} {\vec{r}}_{ε}^{A} \cdot {\underline{u}}_{ε} (t) d x | \\ ⩽ C ε^{M + \frac{1}{2}} ({‖ {\underline{u}}_{ε} (t) ‖}_{L^{2} {(Ω)}^{m}} + {‖ \nabla_{τ} {\underline{u}}_{ε} (t) ‖}_{L^{2} {(Γ_{t} (δ))}^{N \times m}} + {‖ \nabla {\underline{u}}_{ε} (t) ‖}_{L^{2} {(Ω ∖ Γ_{t} (δ))}^{N \times m}}) \end{matrix} \end{matrix}$ for all $ε \in (0, {\overset{ˇ}{ε}}_{0}]$ and $t \in (0, T_{0}]$ .

Well-Prepared Initial Data: For all $ε \in (0, {\overset{ˇ}{ε}}_{0}]$ it holds $\begin{matrix} (6.19) & {‖ {\vec{u}}_{0, ε} - {\vec{u}}_{ε}^{A} |_{t = 0} ‖}_{L^{2} {(Ω)}^{m}} ⩽ R ε^{M + \frac{1}{2}} . \end{matrix}$

Then we obtain

Let $M > k (N)$ . Then there are $β, {\overset{ˇ}{ε}}_{1} > 0$ such that for $g_{β} (t) : = e^{- β t}$ it holds $\begin{matrix} (6.20) & \begin{matrix} sup_{t \in [0, T]} {‖ (g_{β} {\underline{u}}_{ε}) (t) ‖}_{L^{2} {(Ω)}^{m}}^{2} + ‖ g_{β} \nabla {\underline{u}}_{ε} ‖_{L^{2} {(Q_{T} ∖ Γ (δ))}^{N \times m}}^{2} ⩽ 2 R^{2} ε^{2 M + 1}, \\ {\overset{ˇ}{c}}_{0} ‖ g_{β} \nabla_{τ} {\underline{u}}_{ε} ‖_{L^{2} {(Q_{T} \cap Γ (δ))}^{N \times m}}^{2} + ε^{2} ‖ g_{β} \partial_{n} {\underline{u}}_{ε} ‖_{L^{2} {(Q_{T} \cap Γ (δ))}^{m}}^{2} ⩽ 2 R^{2} ε^{2 M + 1} \end{matrix} \end{matrix}$ for all $ε \in (0, {\overset{ˇ}{ε}}_{1}]$ and $T \in (0, T_{0}]$ .

Let $k (N) \in N$ and $M = k (N)$ . Let ( 6.18 ) hold for some $\overset{ˇ}{M} > M$ instead of M. Then there are $β, \overset{ˇ}{R}, {\overset{ˇ}{ε}}_{1} > 0$ such that, if ( 6.19 ) holds for $\overset{ˇ}{R}$ instead of R, then ( 6.20 ) for $\overset{ˇ}{R}$ instead of R is valid for all $ε \in (0, {\overset{ˇ}{ε}}_{1}], T \in (0, T_{0}]$ .

Let $N \in {2, 3}$ and $M = 2 (= k (N))$ . Then there are ${\overset{ˇ}{ε}}_{1}, {\overset{ˇ}{T}}_{1} > 0$ such that ( 6.20 ) holds for $β = 0$ and for all $ε \in (0, {\overset{ˇ}{ε}}_{1}]$ , $T \in (0, {\overset{ˇ}{T}}_{1}]$ .

Remark 6.10.

The parameter M corresponds to the order of the approximate solution constructed in Section 5.2.

The comments for the scalar case in Remark 6.6, 2.–4., on the role of the parameters β, $k (N)$ and weaker requirements in the theorem, hold analogously for the vector-valued case. More precisely, see (6.26) below. The critical order $k (N)$ is the same as in the scalar case because we use the same Gagliardo–Nirenberg estimates that were used in the scalar case in the proof of Lemma 6.7.

Proof of Theorem 6.9.
The continuity of the objects on the left hand side in (6.20) yields that $\begin{matrix} (6.21) & {\overset{ˇ}{T}}_{ε, β, R} : = sup {\tilde{T} \in (0, T_{0}] : (6.20) holds for ε, R and all T \in (0, \tilde{T}]} \end{matrix}$ is well-defined for all $ε \in (0, {\overset{ˇ}{ε}}_{0}], β ⩾ 0$ and ${\overset{ˇ}{T}}_{ε, β, R} > 0$ . In the different cases we have to show:
If $M > k (N)$ , then there are $β, {\overset{ˇ}{ε}}_{1} > 0$ such that ${\overset{ˇ}{T}}_{ε, β, R} = T_{0}$ for all $ε \in (0, {\overset{ˇ}{ε}}_{1}]$ .

If $M = k (N) \in N$ , then there are $β, \overset{ˇ}{R}, {\overset{ˇ}{ε}}_{1} > 0$ such that ${\overset{ˇ}{T}}_{ε, β, \overset{ˇ}{R}} = T_{0}$ provided that $ε \in (0, {\overset{ˇ}{ε}}_{1}]$ and (6.18) is true for some $\overset{ˇ}{M} > M$ instead of M and (6.19) is valid with R replaced by $\overset{ˇ}{R}$ .

If $N \in {2, 3}$ , $M = 2$ , then there are ${\overset{ˇ}{T}}_{1}, {\overset{ˇ}{ε}}_{1} > 0$ such that ${\overset{ˇ}{T}}_{ε, 0, R} ⩾ {\overset{ˇ}{T}}_{1}$ for all $ε \in (0, {\overset{ˇ}{ε}}_{1}]$ .

We do a general computation first and consider the specific cases later. The difference of the left hand sides in (vAC1) for ${\vec{u}}_{ε}$ and ${\vec{u}}_{ε}^{A}$ yields $\begin{matrix} (6.22) & [\partial_{t} - Δ + \frac{1}{ε^{2}} D^{2} W ({\vec{u}}_{ε}^{A})] {\underline{u}}_{ε} = - {\vec{r}}_{ε}^{A} - {\vec{r}}_{ε} ({\vec{u}}_{ε}, {\vec{u}}_{ε}^{A}), \end{matrix}$ where ${\vec{r}}_{ε} ({\vec{u}}_{ε}, {\vec{u}}_{ε}^{A}) : = \frac{1}{ε^{2}} [\nabla W ({\vec{u}}_{ε}) - \nabla W ({\vec{u}}_{ε}^{A}) - D^{2} W ({\vec{u}}_{ε}^{A}) {\underline{u}}_{ε}]$ . We multiply (6.22) by $g_{β}^{2} {\underline{u}}_{ε}$ and integrate over $Q_{T}$ for $T \in (0, {\overset{ˇ}{T}}_{ε, β, R}]$ , where $ε \in (0, {\overset{ˇ}{ε}}_{0}]$ and $β ⩾ 0$ are fixed. This implies $\begin{matrix} (6.23) & \begin{matrix} \int_{0}^{T} g_{β}^{2} \int_{Ω} {\underline{u}}_{ε} \cdot [\partial_{t} - Δ + \frac{1}{ε^{2}} D^{2} W ({\vec{u}}_{ε}^{A})] {\underline{u}}_{ε} d x d t \\ = - \int_{0}^{T} g_{β}^{2} \int_{Ω} [{\vec{r}}_{ε}^{A} + {\vec{r}}_{ε} ({\vec{u}}_{ε}, {\vec{u}}_{ε}^{A})] \cdot {\underline{u}}_{ε} d x d t \end{matrix} \end{matrix}$ for all $T \in (0, {\overset{ˇ}{T}}_{ε, β, R}]$ , $ε \in (0, {\overset{ˇ}{ε}}_{0}]$ and $β ⩾ 0$ . We estimate all terms. First, $\frac{1}{2} \partial_{t} | {\underline{u}}_{ε} |^{2} = {\underline{u}}_{ε} \cdot \partial_{t} {\underline{u}}_{ε}$ , integration by parts in time and $\partial_{t} g_{β} = - β g_{β}$ yield $\begin{matrix} \int_{0}^{T} \int_{Ω} g_{β}^{2} \partial_{t} {\underline{u}}_{ε} \cdot {\underline{u}}_{ε} d x d t = \frac{1}{2} g_{β} {(T)}^{2} {‖ {\underline{u}}_{ε} (T) ‖}_{L^{2} {(Ω)}^{m}}^{2} - \frac{1}{2} {‖ {\underline{u}}_{ε} (0) ‖}_{L^{2} {(Ω)}^{m}}^{2} + β \int_{0}^{T} g_{β}^{2} ‖ {\underline{u}}_{ε} ‖_{L^{2} {(Ω)}^{m}}^{2} d t, \end{matrix}$ where $‖ {\underline{u}}_{ε} (0) ‖_{L^{2} {(Ω)}^{m}}^{2} ⩽ R^{2} ε^{2 M + 1}$ because of (6.19) (“well-prepared initial data”). For the other term on the left hand side in (6.23) we use integration by parts in space. This yields $\begin{array}{l} \int_{0}^{T} g_{β}^{2} \int_{Ω} {\underline{u}}_{ε} \cdot [- Δ + \frac{1}{ε^{2}} D^{2} W ({\vec{u}}_{ε}^{A})] {\underline{u}}_{ε} d x d t \\ = \int_{0}^{T} g_{β}^{2} \int_{Ω} | \nabla {\underline{u}}_{ε} |^{2} + \frac{1}{ε^{2}} {({\underline{u}}_{ε}, D^{2} W ({\vec{u}}_{ε}^{A} (\cdot, t)) {\underline{u}}_{ε})}_{R^{m}} d x d t + \int_{0}^{T} g_{β}^{2} \int_{\partial Ω} {\vec{s}}_{ε}^{A} \cdot tr {\underline{u}}_{ε} d H^{N - 1} d t . \end{array}$ Using requirement 2. (“spectral estimate”) in the theorem we obtain that the first integral on the right hand side of the latter equation is bounded from below by $\begin{matrix} - \overset{ˇ}{C} ‖ {\underline{u}}_{ε} ‖_{L^{2} {(Ω)}^{m}}^{2} + ‖ \nabla {\underline{u}}_{ε} ‖_{L^{2} {(Ω ∖ Γ_{t} (δ))}^{N \times m}}^{2} + {\overset{ˇ}{c}}_{0} ‖ \nabla_{τ} {\underline{u}}_{ε} ‖_{L^{2} {(Γ_{t} (δ))}^{N \times m}}^{2} . \end{matrix}$ For the remainder terms involving ${\vec{r}}_{ε}^{A}$ and ${\vec{s}}_{ε}^{A}$ we apply (6.18) (“approximate solution”). Hence $\begin{matrix} | \int_{0}^{T} g_{β}^{2} [\int_{\partial Ω} {\vec{s}}_{ε}^{A} \cdot tr {\underline{u}}_{ε} (t) d H^{N - 1} + \int_{Ω} {\vec{r}}_{ε}^{A} \cdot {\underline{u}}_{ε} (t) d x] d t | ⩽ {\overset{ˇ}{C}}_{1} R ‖ g_{β} ‖_{L^{2} (0, T)} ε^{2 M + 1} \end{matrix}$ due to (6.20) for all $T \in (0, {\overset{ˇ}{T}}_{ε, β, R}]$ , $ε \in (0, {\overset{ˇ}{ε}}_{0}]$ , where $‖ g_{β} ‖_{L^{1} (0, T)} ⩽ \sqrt{T_{0}} ‖ g_{β} ‖_{L^{2} (0, T)}$ is used.

Now we estimate the ${\vec{r}}_{ε}$ -term in (6.23). The requirement 1. (“uniform boundedness”) in the theorem and Lemma 6.2 yield $\begin{matrix} (6.24) & sup_{ε \in (0, {\overset{ˇ}{ε}}_{0}]} [‖ {\vec{u}}_{ε} ‖_{L^{\infty} {(Q_{T_{0}})}^{m}} + {‖ {\vec{u}}_{ε}^{A} ‖}_{L^{\infty} {(Q_{T_{0}})}^{m}}] < \infty . \end{matrix}$ Therefore the Taylor Theorem yields $\begin{matrix} (6.25) & | \int_{0}^{T} g_{β}^{2} \int_{Ω} {\vec{r}}_{ε} ({\vec{u}}_{ε}, {\vec{u}}_{ε}^{A}) \cdot {\underline{u}}_{ε} d x d t | ⩽ \frac{C}{ε^{2}} \int_{0}^{T} g_{β}^{2} ‖ {\underline{u}}_{ε} ‖_{L^{3} {(Ω)}^{m}}^{3} d t . \end{matrix}$ This term can be estimated in the analogous way as in the scalar case with Gagliardo–Nirenberg inequalities on $Ω ∖ Γ_{t} (δ)$ and $Γ_{t} (δ)$ , cf. the proof of Lemma 6.7. This yields $\begin{matrix} | \int_{0}^{T} g_{β}^{2} \int_{Ω} {\vec{r}}_{ε} ({\vec{u}}_{ε}, {\vec{u}}_{ε}^{A}) \cdot {\underline{u}}_{ε} d x d t | ⩽ C R^{2 + K (N)} ε^{2 M + 1} ε^{K (N) (M - k (N))} {‖ g_{β}^{- K (N)} ‖}_{L^{\frac{4}{4 - min {4, N}}} (0, T)} \end{matrix}$ for all $T \in (0, {\overset{ˇ}{T}}_{ε, β, R}]$ and $ε \in (0, {\overset{ˇ}{ε}}_{0}]$ , where $K (N) : = min {1, \frac{4}{N}} \in (0, 1]$ .

In order to control $\partial_{n} {\underline{u}}_{ε}$ we use Corollary 3.5 and obtain $\begin{array}{l} ε^{2} ‖ g_{β} \partial_{n} {\underline{u}}_{ε} ‖_{L^{2} {(Q_{T} \cap Γ (δ))}^{m}}^{2} & ⩽ C ε^{2} \int_{0}^{T} g_{β}^{2} \int_{Ω} | \nabla {\underline{u}}_{ε} |^{2} + \frac{1}{ε^{2}} {({\underline{u}}_{ε}, D^{2} W ({\vec{u}}_{ε}^{A} (\cdot, t)) {\underline{u}}_{ε})}_{R^{m}} d x d t \\ + C sup_{ε \in (0, {\overset{ˇ}{ε}}_{0}]} {‖ D^{2} W ({\vec{u}}_{ε}^{A}) ‖}_{L^{\infty} {(Q_{T_{0}})}^{m \times m}} \int_{0}^{T} g_{β}^{2} {‖ {\underline{u}}_{ε} (t) ‖}_{L^{2} {(Ω)}^{m}}^{2} d t \end{array}$ with a constant $C > 0$ independent of ε, T and R. The first term is absorbed with $\frac{1}{2}$ of the spectral term above if $ε \in (0, {\overset{ˇ}{ε}}_{1}]$ and ${\overset{ˇ}{ε}}_{1} > 0$ is small (independent of T, R). Altogether we obtain $\begin{matrix} (6.26) & \begin{matrix} \frac{1}{2} g_{β} (T) {‖ {\underline{u}}_{ε} (T) ‖}_{L^{2} {(Ω)}^{m}}^{2} + \frac{1}{2} ‖ g_{β} \nabla {\underline{u}}_{ε} ‖_{L^{2} {(Q_{T} ∖ Γ (δ))}^{N \times m}}^{2} \\ + \frac{{\overset{ˇ}{c}}_{0}}{2} ‖ g_{β} \nabla_{τ} {\underline{u}}_{ε} ‖_{L^{2} {(Q_{T} \cap Γ (δ))}^{N \times m}}^{2} + \frac{1}{2} ε^{2} ‖ g_{β} \partial_{n} {\underline{u}}_{ε} ‖_{L^{2} {(Q_{T} \cap Γ (δ))}^{m}}^{2} \\ ⩽ \frac{R^{2}}{2} ε^{2 M + 1} + \int_{0}^{T} (- β + {\overset{ˇ}{C}}_{0}) g_{β}^{2} {‖ {\underline{u}}_{ε} (t) ‖}_{L^{2} {(Ω)}^{m}}^{2} d t + {\overset{ˇ}{C}}_{1} R ε^{2 M + 1} ‖ g_{β} ‖_{L^{2} (0, T)} \\ + C R^{2 + K (N)} ε^{2 M + 1} ε^{K (N) (M - k (N))} {‖ g_{β}^{- K (N)} ‖}_{L^{\frac{4}{4 - min {4, N}}} (0, T)} \end{matrix} \end{matrix}$ for all $T \in (0, {\overset{ˇ}{T}}_{ε, β, R}]$ , $ε \in (0, {\overset{ˇ}{ε}}_{1}]$ and constants ${\overset{ˇ}{C}}_{0}, {\overset{ˇ}{C}}_{1}, C > 0$ independent of $ε, T, R$ , where $k (N) = max {2, \frac{N}{2}}$ and $K (N) = min {1, \frac{4}{N}}$ .

Now we consider the different cases in the theorem.
Ad 1.
If $M > k (N)$ , then we choose $β ⩾ {\overset{ˇ}{C}}_{0}$ large such that ${\overset{ˇ}{C}}_{1} R ‖ g_{β} ‖_{L^{2} (0, T_{0})} ⩽ \frac{R^{2}}{8}$ . Then (6.26) is estimated by $\frac{3}{4} R^{2} ε^{2 M + 1}$ for all $T \in (0, {\overset{ˇ}{T}}_{ε, β, R}]$ and $ε \in (0, {\overset{ˇ}{ε}}_{1}]$ , if ${\overset{ˇ}{ε}}_{1} > 0$ is small. Via contradiction and continuity this proves ${\overset{ˇ}{T}}_{ε, β, R} = T_{0}$ for all $ε \in (0, {\overset{ˇ}{ε}}_{1}]$ . □
Ad 2.
Let $M = k (N) \in N$ and let (6.18) hold for some $\overset{ˇ}{M} > M$ instead of M. Then the term in (6.26) where R enters linearly is improved by a factor $ε^{\overset{ˇ}{M} - M}$ . Let $β ⩾ {\overset{ˇ}{C}}_{0}$ be fixed. Now we first choose $R > 0$ small such that the $R^{2 + K (N)}$ -term in (6.26) is estimated by $\frac{1}{8} R^{2} ε^{2 M + 1}$ . Then ${\overset{ˇ}{ε}}_{1} > 0$ can be taken small such that (6.26) is bounded by $\frac{3}{4} R^{2} ε^{2 M + 1}$ for all $T \in (0, {\overset{ˇ}{T}}_{ε, β, R}]$ and $ε \in (0, {\overset{ˇ}{ε}}_{1}]$ . By contradiction and continuity we get ${\overset{ˇ}{T}}_{ε, β, R} = T_{0}$ for all $ε \in (0, {\overset{ˇ}{ε}}_{1}]$ . □
Ad 3.
Finally, let $N \in {2, 3}$ , $M = 2$ and $β = 0$ . Then (6.26) is estimated by $\begin{matrix} [\frac{R^{2}}{2} + C R^{2} T + C R T^{\frac{1}{2}} + C R^{3} T^{\frac{4 - N}{4}}] ε^{2 M + 1} . \end{matrix}$ Due to $\frac{4 - N}{4} > 0$ there are ${\overset{ˇ}{ε}}_{1}, {\overset{ˇ}{T}}_{1} > 0$ such that the latter is bounded by $\frac{3}{4} R^{2} ε^{2 M + 1}$ for every $T \in (0, min (T_{ε, β, R}, {\overset{ˇ}{T}}_{1})]$ and $ε \in (0, {\overset{ˇ}{ε}}_{1}]$ . Therefore ${\overset{ˇ}{T}}_{ε, 0, R} ⩾ {\overset{ˇ}{T}}_{1}$ for all $ε \in (0, {\overset{ˇ}{ε}}_{1}]$ . □

The proof of Theorem 6.9 is completed. □
6.3.2. Proof of Theorem 1.6

Let $N ⩾ 2$ , Ω, $Q_{T}$ and $\partial Q_{T}$ be as in Remark 1.3, 1. Let $W : R^{m} \to R$ be as in Definition 1.1 and ${\vec{u}}_{\pm}$ be any distinct pair of minimizers of W. Moreover, let $Γ = {(Γ_{t})}_{t \in [0, T_{0}]}$ for some $T_{0} > 0$ be a smooth solution to (MCF) with 90°-contact angle condition parametrized as in Section 3.1 and let $δ > 0$ be such that Theorem 3.2 holds for $2 δ$ instead of δ. We use the notation from Section 3.1 and Section 3.2. Let $M \in N$ with $M ⩾ 2$ and denote with ${({\vec{u}}_{ε}^{A})}_{ε > 0}$ the approximate solution on $\overline{Q_{T_{0}}}$ from Section 5.2.3 (that was constructed with asymptotic expansions in Section 5.2) and let ${\overset{ˇ}{ε}}_{0} > 0$ be such that Lemma 5.16 (“remainder estimate”) holds for $ε \in (0, {\overset{ˇ}{ε}}_{0}]$ . The property ${lim}_{ε \to 0} {\vec{u}}_{ε}^{A} = {\vec{u}}_{\pm}$ uniformly on compact subsets of $Q_{T_{0}}^{\pm}$ follows from Section 5.2.

Theorem 1.6 follows directly from Theorem 6.9 if we prove the conditions 1.–4. in Theorem 6.9. The requirement 1. (“uniform boundedness”) is satisfied because of Lemma 5.16 for ${\vec{u}}_{ε}^{A}$ and for ${\vec{u}}_{0, ε}$ this is an assumption in Theorem 1.6, Condition 2. (“spectral estimate”) is precisely the assertion in [67, Theorem 1.7]. Requirement 4. (“well prepared initial data”) is a condition on ${\vec{u}}_{0, ε}$ and assumed in Theorem 1.6. It remains to prove 3. (“approximate solution”). This can be done in the analogous way as in the scalar case, cf. the proof of Theorem 1.4 in Section 6.2.2. Basically one uses suitable integral transformations, Hölder estimates, transformation arguments like in [67, Lemma 5.4], the properties of ${\vec{r}}_{ε}^{A}$ , ${\vec{s}}_{ε}^{A}$ from Lemma 5.16 as well as the comparison of several differential operators in Corollary 3.5. Since the computations are completely analogous to the scalar case, we refrain from going into details.

Footnotes

Acknowledgements

The author gratefully acknowledges support through DFG, GRK 1692 “Curvature, Cycles and Cohomology” during parts of the work.

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Convergence of the scalar- and vector-valued Allen–Cahn equation to mean curvature flow with 90°-contact angle in higher dimensions,part I: Convergence result

Abstract

Keywords

1. Introduction

1.1. Result in scalar case

Theorem 1.4 (Convergence of (AC) to (MCF) with 90°-Contact Angle).

Theorem 1.6 (Convergence of (vAC) to (MCF) with 90°-Contact Angle).

2 For convenience. The considerations can be adapted for the case of finitely many connected components.

Theorem 3.2 (Curvilinear Coordinates).

4.1. Some scalar-valued ODE problems on R

4.1.1. The ODE for the optimal profile

Theorem 4.1 ([75, Lemma 2.6.1]).

Theorem 4.2 ([75, Lemma 2.6.2]).

4.2.1. Weak solutions and regularity

Definition 4.3 ([5, Definition 2.7]).

Theorem 4.4 ([5, Theorem 2.9]).

Corollary 4.5 ([5, Corollary 2.10]).

4.2.2. Solution operators in exponentially weighted spaces

Theorem 4.6 (Solution Operators for Decay inH).

4.3.1. The nonlinear ODE

Theorem 4.9 ([54, Section 2.1], [22, Section 2]).

Remark 4.11 (Assumption dim ker L ˇ 0 = 1 ).

4.4.1. Weak solutions and regularity

Theorem 4.18 (Solution Operators for the Vector-valued Case).

5. Asymptotic expansions

5.1. Asymptotic expansion of (AC) in ND

5.1.1. Inner expansion of (AC) in ND

4 Note that this set only appears in the contact point expansion later.

5.2.1. Inner expansion of (vAC) in ND

5.2.2. Contact point expansion of (vAC) in ND

5.2.3. The approximate solution for (vAC) in ND

6.1. Preliminaries

6.1.1. Uniform a priori bound for classical solutions of (AC)

Lemma 6.3 (Gagliardo–Nirenberg Inequality).

6.2.1. Difference estimate

Theorem 6.5 (Difference Estimate for (AC)).

6.3. Difference estimate and proof of the convergence theorem for (vAC) in ND

6.3.1. Difference estimate

Theorem 6.9 (Difference Estimate for (vAC)).

Footnotes

Acknowledgements

References

²
For convenience. The considerations can be adapted for the case of finitely many connected components.

4.1. Some scalar-valued ODE problems on $R$

Remark 4.11 (Assumption $dim ker {\overset{ˇ}{L}}_{0} = 1$ ).

⁴
Note that this set only appears in the contact point expansion later.