A note on a phase-field model for anisotropic systems

Abstract

We investigate, using the framework of Γ-convergence, a phase-field model proposed in [Proc. R. Soc. A 465 (2009), 1337–1359] for strongly anisotropic systems; in particular, we prove a full Γ-convergence result for an anisotropic Modica–Mortola-type energy.

Keywords

phase-field anisotropy Γ-convergence

1. Introduction

In [13] a new phase-field model for strongly anisotropic crystals and epitaxial growth has been proposed using anisotropic Cahn–Hilliard-type equations regularized by an high-order Willmore term. More precisely, the authors introduced the energy functional $F_{ε} (u) = \frac{1}{ε} \int_{Ω} γ (\frac{\nabla u}{| \nabla u |}) (\frac{ε^{2}}{2} | \nabla u |^{2} + W (u)) d x + \frac{β}{2 ε} \int_{Ω} {(- ε Δ u + \frac{1}{ε} W^{'} (u))}^{2} d x,$ where Ω is a bounded and open subset of $R^{n}$ , $u : Ω \to R$ represents the phase in a multiphase system, $γ : {x \in R^{n} : | x | = 1} \to (0, + \infty)$ is a Lipschitz continuous function, W is a double well potential, with wells in 0 and 1, β is a fixed positive constant and ε is the measure of the transition layer thickness. In [13] the authors said that $F_{ε}$ approaches as $ε \to 0$ , a limit energy of the form $\int_{S} γ (n) d S + \frac{β}{2} \int_{S} H_{S}^{2} d S,$ (1.1) where S is the limit sharp interface created by a sequence $u_{ε}$ which makes the energy equibounded, n is a unit normal to S which comes from the diffuse normal $\nabla u / | \nabla u |$ , and $H_{S}$ is the mean curvature at S, term that arises from the second order penalization that appears in $F_{ε}$ . Notice that the limit functional (1.1) is composed by two terms: the first one represents an anisotropic perimeter, which is an interesting object recently investigated in view of other applications (see for instance [5] and [6]), while the second one turns out to be, up to a constant, the well-known Willmore functional. A rigorous analysis of $F_{ε}$ using the framework of Γ-convergence (see [4] and [7] for details) reduces to the well-known Modica–Mortola setting (see for instance [9] and [10]) when $γ = 1$ and $β = 0$ , and precisely if $F_{ε} (u_{ε})$ is bounded then $u_{ε}$ converges, up to subsequences, to some $u \in BV (Ω; {0, 1})$ strongly in $L^{1} (Ω)$ and the Γ-limit of $F_{ε}$ is given by $c_{0} H^{n - 1} (J_{u})$ where $c_{0}$ is a suitable positive constant depending only on W and $H^{n - 1}$ is the $(n - 1)$ -dimensional Hausdorff measure. First of all, in this paper we will investigate the behavior of the first part of $F_{ε}$ ; this could be interesting from a mathematical point of view even without the second order penalization. We are able to prove a complete Γ-convergence result, and roughly speaking it turns out that $\frac{1}{ε} \int_{Ω} γ (\frac{\nabla u}{| \nabla u |}) (\frac{ε^{2}}{2} | \nabla u |^{2} + W (u)) d x \overset{Γ}{\to} \int_{S} \bar{γ} (n) d S,$ where $\bar{γ}$ is the convexification of the positively one-homogeneous extension of γ. The situation is much more complicated if we try to compute the Γ-limit of the full energy: in this case, if $γ = β = 1$ it turns out that if $F_{ε} (u_{ε})$ is bounded then $u_{ε}$ converges, up to subsequences, to some $u \in BV (Ω; {0, 1})$ strongly in $L^{1} (Ω)$ and the Γ-limit of $F_{ε}$ restricted to sufficiently smooth sets $E \subseteq R^{n}$ (boundary of class $C^{2}$ for instance is enough) is given by $c_{0} H^{n - 1} (\partial E \cap Ω) + c_{0} \int_{\partial E \cap Ω} H_{\partial E \cap Ω}^{2} d H^{n - 1} .$ More precisely, the Γ-lim sup inequality always holds true (see [3]) while the Γ-lim inf estimate can be proved only for dimensions 2 and 3 (see [11] and [12]). In the last section of this paper we try to prove a Γ-convergence result for the energy $F_{ε}$ when $γ \neq 1$ , but unfortunately we get only partial results: we are only able to prove, and actually this will be almost straightforward, that the functional $\int_{S} γ (n) d S + \frac{β}{2} \int_{S} H_{S}^{2} d S$ is the Γ-limit of $F_{ε}$ only in dimension 2 and 3, on sufficiently smooth interfaces, and also under a suitable convexity assumption on γ.

2. Some preliminary considerations

We now move to a more precise setting, and we look better the first part of the energy, that is $\frac{1}{ε} \int_{Ω} γ (\frac{\nabla u}{| \nabla u |}) (\frac{ε^{2}}{2} | \nabla u |^{2} + W (u)) d x,$ (2.1) where the anisotropy function $γ : S^{n - 1} \to (0, + \infty)$ is Lipschitz continuous with Lipschitz constant $L > 0$ , and $S^{n - 1} : = {x \in R^{n} : | x | = 1}$ . The first obvious remark we can do is that the expression (2.1) does not make sense if $\nabla u = 0$ ; in order to avoid this problem, and to have an expression which is always well defined, we simply extend γ by its positively one-homogeneous extension $γ_{0} : R^{n} \to [0, + \infty)$ given by $γ_{0} (ξ) : = \{\begin{matrix} γ (\frac{ξ}{| ξ |}) | ξ | & if ξ \neq 0, \\ 0 & if ξ = 0 \end{matrix}$ and instead of $γ (\nabla u / | \nabla u |)$ we take $γ_{0} (\frac{\nabla u}{\sqrt{r^{2} + | \nabla u |^{2}}}) for r small .$ Notice that when $r \sim 0$ and $\nabla u \neq 0$ we get, since $γ_{0}$ is positively one-homogeneous, $γ_{0} (\frac{\nabla u}{\sqrt{r^{2} + | \nabla u |^{2}}}) = γ_{0} (\frac{\nabla u}{| \nabla u |}) \frac{| \nabla u |}{\sqrt{r^{2} + | \nabla u |^{2}}} \sim γ_{0} (\frac{\nabla u}{| \nabla u |}) = γ (\frac{\nabla u}{| \nabla u |}) .$ Therefore, the first idea is to replace the functional in (2.1) by the new one $\frac{1}{ε} \int_{Ω} γ_{0} (\frac{\nabla u}{\sqrt{r_{ε}^{2} + | \nabla u |^{2}}}) (\frac{ε^{2}}{2} | \nabla u |^{2} + W (u)) d x,$ (2.2) where now $r_{ε} \to 0$ as $ε \to 0$ ; we will more precise later about the speed of convergence of $r_{ε}$ to 0. In any case, the Γ-limit of (2.2) should be finite only on $BV (Ω; {0, 1})$ , as in the classical Modica–Mortola case where simply $γ = 1$ , and we expect that it looks like an anisotropic perimeter of the form $\int_{J_{u}} ϕ (ν_{u}) d H^{n - 1}$ for some anisotropy ϕ related to γ. But, notice that if $u_{ε} = c$ for some c constant we have $lim_{ε \to 0} \frac{1}{ε} \int_{Ω} γ_{0} (\frac{\nabla u_{ε}}{\sqrt{r^{2} + | \nabla u_{ε} |^{2}}}) (\frac{ε^{2}}{2} | \nabla u_{ε} |^{2} + W (u_{ε})) d x = 0$ which says that the Γ-limit of (2.2) is 0 on constant functions. We therefore need to prevent $γ_{0} = 0$ without changing $γ_{0}$ on $S^{n - 1}$ . In order to do this, let $B_{R} (0)$ be the ball centered in 0 with radius $R > 0$ , take a cut-off function $φ : R^{n} \to [0, 1]$ with $φ = 1$ on $R^{n} ∖ B_{1 / 2} (0)$ , with $φ = 0$ on $B_{1 / 4} (0)$ , and with $| \nabla φ | ⩽ 4$ everywhere, and let $\tilde{γ} : = φ γ_{0} + (1 - φ) \frac{m}{4}$ , where $m : = min γ$ . Notice that $\tilde{γ} = γ_{0}$ on $\partial B_{1} (0) = S^{n - 1}$ . Moreover, since for any $ξ \in \overline{B_{1} (0)}$ with $| ξ | ⩾ 1 / 4$ we have $γ_{0} (ξ) ⩾ m / 4$ , it holds $\tilde{γ} ⩾ m / 4$ , which is what we wanted.

Proposition 2.1.

The function $\tilde{γ}$ is Lipschitz continuous.

Proof.

In order to prove that $\tilde{γ}$ is Lipschitz continuous it is sufficient to prove that $γ_{0}$ is Lipschitz continuous. The Lipschitzianity of $γ_{0}$ easily descends from the elementary inequality $\sqrt{| ξ_{1} |} \sqrt{| ξ_{2} |} | \frac{ξ_{1}}{| ξ_{1} |} - \frac{ξ_{2}}{| ξ_{2} |} | ⩽ | ξ_{1} - ξ_{2} |, ξ_{1}, ξ_{2} \neq 0$ (2.3) which can be proved as follows: if $α : = \sqrt{| ξ_{1} | / | ξ_{2} |}$ then $\frac{| ξ_{1} - ξ_{2} |}{\sqrt{| ξ_{1} |} \sqrt{| ξ_{2} |}} = | α \frac{ξ_{1}}{| ξ_{1} |} - \frac{1}{α} \frac{ξ_{2}}{| ξ_{2} |} | = \sqrt{α^{2} + \frac{1}{α^{2}} - 2 \frac{ξ_{1}}{| ξ_{1} |} \cdot \frac{ξ_{2}}{| ξ_{2} |}} ⩾ \sqrt{2 - 2 \frac{ξ_{1}}{| ξ_{1} |} \cdot \frac{ξ_{2}}{| ξ_{2} |}} = | \frac{ξ_{1}}{| ξ_{1} |} - \frac{ξ_{2}}{| ξ_{2} |} | .$ Let now $ξ_{1}, ξ_{2} \in R^{n}$ with $ξ_{i} \neq 0$ for $i = 1, 2$ (the case $ξ_{1} = 0$ or $ξ_{2} = 0$ is trivial). Without loss of generality we can assume $| ξ_{1} | ⩽ | ξ_{2} |$ . Applying (2.3) we obtain $\begin{array}{rcl} | γ_{0} (ξ_{1}) - γ_{0} (ξ_{2}) | & = & | γ (\frac{ξ_{1}}{| ξ_{1} |}) | ξ_{1} | - γ (\frac{ξ_{2}}{| ξ_{2} |}) | ξ_{2} | | \\ = & | γ (\frac{ξ_{1}}{| ξ_{1} |}) | ξ_{1} | - γ (\frac{ξ_{2}}{| ξ_{2} |}) | ξ_{2} - ξ_{1} + ξ_{1} | | \\ ⩽ & | γ (\frac{ξ_{1}}{| ξ_{1} |}) | ξ_{1} | - γ (\frac{ξ_{2}}{| ξ_{2} |}) | ξ_{1} | | + max γ | ξ_{1} - ξ_{2} | \\ ⩽ & L | ξ_{1} | | \frac{ξ_{1}}{| ξ_{1} |} - \frac{ξ_{2}}{| ξ_{2} |} | + max γ | ξ_{1} - ξ_{2} | \\ ⩽ & L \sqrt{| ξ_{1} |} \sqrt{| ξ_{2} |} | \frac{ξ_{1}}{| ξ_{1} |} - \frac{ξ_{2}}{| ξ_{2} |} | + max γ | ξ_{1} - ξ_{2} | \\ ⩽ & (L + max γ) | ξ_{1} - ξ_{2} | \end{array}$ and this concludes the proof. □

We have therefore obtained a Lipschitz continuous function $\tilde{γ} : R^{n} \to (0, + \infty)$ , and the idea is to consider the energy functional $\frac{1}{ε} \int_{Ω} \tilde{γ} (\frac{\nabla u}{\sqrt{r^{2} + | \nabla u |^{2}}}) (\frac{ε^{2}}{2} | \nabla u |^{2} + W (u)) d x$ which will be studied in the next section.

3. Analysis of the anisotropic Modica–Mortola energy

In this section we investigate the behavior of the first part of the energy in a general setting.

3.1. The anisotropy

Let $γ : R^{n} \to (0, + \infty)$ be a Lipschitz continuous function with Lipschitz constant $L > 0$ and with $γ (ξ) = γ (- ξ)$ for each $ξ \in S^{n - 1}$ . For any $r > 0$ let $γ_{r} : R^{n} \to (0, + \infty)$ be given by $γ_{r} (ξ) : = γ (\frac{ξ}{\sqrt{r^{2} + | ξ |^{2}}}) .$ For any $ξ \in R^{n}$ let $γ_{0} (ξ) : = \{\begin{matrix} γ (\frac{ξ}{| ξ |}) | ξ | & if ξ \neq 0, \\ 0 & if ξ = 0 . \end{matrix}$ For any $r > 0$ and for any $ξ \in R^{n}$ with $ξ \neq 0$ we have $| γ (\frac{ξ}{\sqrt{r^{2} + | ξ |^{2}}}) - γ (\frac{ξ}{| ξ |}) | ⩽ L | \frac{ξ}{\sqrt{r^{2} + | ξ |^{2}}} - \frac{ξ}{| ξ |} | ⩽ \frac{L r}{2 | ξ |}$ from which, multiplying both sides by $| ξ |$ , we get the following useful estimate: $| γ_{r} (ξ) | ξ | - γ_{0} (ξ) | ⩽ L r / 2 .$ (3.1)

3.2. Statement of the convergence result

Let $W : R \to [0, + \infty)$ be the double-well potential given by $W (t) : = t^{2} {(1 - t)}^{2} / 4$ . For each $ε > 0$ let $r_{ε} = O (ε)$ . We define the energy $G_{ε} : L^{1} (Ω) \to [0, + \infty]$ given by $G_{ε} (u) : = \{\begin{matrix} \frac{1}{ε} \int_{Ω} γ_{r_{ε}} (\nabla u) (\frac{ε^{2}}{2} | \nabla u |^{2} + W (u)) d x & if u \in H^{1} (Ω), \\ + \infty & otherwise in L^{1} (Ω) . \end{matrix}$ We are able to prove a full Γ-convergence result for $G_{ε}$ .

Theorem 3.1.
The family $G_{ε}$ Γ-converges as $ε \to 0$ , strongly in $L^{1} (Ω)$ , to $G (u) : = \{\begin{matrix} c_{0} \int_{J_{u}} γ_{0}^{* } (ν_{u}) d H^{n - 1} & if u \in BV (Ω; {0, 1}), \\ + \infty & otherwise in L^{1} (Ω), \end{matrix}$ where* $γ_{0}^{* }$ is the convex envelope of* $γ_{0}$ , $ν_{u}$ is a unit normal at $J_{u}$ and $c_{0} : = \int_{0}^{1} \sqrt{2 W (s)} d s$ .

3.3. The lower estimate

We first recall a relaxation result (see Theorem 1.1 in [8]).

Theorem 3.2.
Assume that $f : Ω \times R \times R^{n} \to [0, + \infty)$ is a Borel integrand, $f (x, u, \cdot)$ is convex in $R^{n}$ , and for all $(x_{0}, u_{0}) \in Ω \times R$ and $η > 0$ there exists $δ > 0$ such that $f (x_{0}, u_{0}, ξ) - f (x, u, ξ) ⩽ η (1 + f (x, u, ξ))$ for all $(x, u) \in Ω \times R$ with $| x - x_{0} | + | u - u_{0} | ⩽ δ$ and for all $ξ \in R^{n}$ . Let $u \in {BV}_{loc} (Ω)$ and let $u_{j} \to u$ strongly in $L_{loc}^{1} (Ω)$ , with $u_{j} \in W_{loc}^{1, 1} (Ω)$ . Then $\begin{array}{rcl} \int_{Ω} f (x, u, \nabla u) d x + \int_{Ω} f^{\infty} (x, u, dC (u)) + \int_{J_{u}} \int_{u^{-} (x)}^{u^{+} (x)} f^{\infty} (x, s, ν_{u}) d s d H^{n - 1} \\ ⩽ \underset{j}{lim inf} \int_{Ω} f (x, u_{j}, \nabla u_{j}) d x, \end{array}$ where $dC (u)$ is the Cantor part of the distributional derivative of u and the so called recession function of f is given by $f^{\infty} (x, u, ξ) : = \underset{t \to + \infty}{lim sup} \frac{f (x, u, t ξ)}{t} .$

In the next proposition we prove the Γ-lim inf inequality for $G_{ε}$ ; for, let us fix a positive infinitesimal sequence $(ε_{j})$ . Proposition 3.3.
Let $u \in L^{1} (Ω)$ and let $(u_{j})$ be a sequence in $H^{1} (Ω)$ with ${sup}_{j} G_{ε_{j}} (u_{j}) < + \infty$ . Then, up to subsequence, not relabeled, we have $u_{j} \to u$ strongly in $L^{1} (Ω)$ for some $u \in BV (Ω; {0, 1})$ . Moreover, for any $u \in BV (Ω; {0, 1})$ and for any sequence $(u_{j})$ in $H^{1} (Ω)$ with $u_{j} \to u$ strongly in $L^{1} (Ω)$ , we have $\underset{j}{lim inf} G_{ε_{j}} (u_{j}) ⩾ G (u) .$ (3.2)
Proof.
First of all, notice that $sup_{j} G_{ε_{j}} (u_{j}) ⩾ \frac{min γ}{ε_{j}} \int_{Ω} (\frac{ε_{j}^{2}}{2} | \nabla u_{j} |^{2} + W (u_{j})) d x .$ Hence, the compactness of the sequence $(u_{j})$ descends from the well-known compactness property of the Modica–Mortola functional (see for instance [9] and [10]). Therefore, in order to prove (3.2) we can assume $G_{ε_{j}} (u_{j}) ⩽ M$ for some $M > 0$ and $u \in BV (Ω; {0, 1})$ . First, notice that by Hölder inequality $\int_{Ω} \sqrt{2 W (u_{j})} d x ⩽ \sqrt{\frac{2 | Ω | M ε_{j}}{min γ}}$ and in particular $\int_{Ω} \sqrt{2 W (u_{j})} d x$ is bounded. Combining Young inequality with (3.1) we obtain, since trivially $γ_{0} ⩾ γ_{0}^{* }$ , $G_{ε_{j}} (u_{j}) ⩾ \int_{Ω} \sqrt{2 W (u_{j})} γ_{r_{ε_{j}}} (\nabla u_{j}) | \nabla u_{j} | d x ⩾ \int_{Ω} \sqrt{W (u_{j})} γ_{0}^{ } (\nabla u_{j}) d x - \frac{r_{ε_{j}} L}{2} \int_{Ω} \sqrt{2 W (u_{j})} d x .$ Since $r_{ε_{j}} \to 0$ we get $\underset{j}{lim inf} G_{ε_{j}} (u_{j}) ⩾ \underset{j}{lim inf} \int_{Ω} \sqrt{W (u_{j})} γ_{0}^{ } (\nabla u_{j}) d x .$ We are now in position to apply Theorem 3.2 with choice $f (x, u, ξ) : = \sqrt{2 W (u)} γ_{0}^{ } (ξ)$ , from which we immediately obtain, since $γ_{0}^{ }$ remains positively one-homogeneous and since the very definition of $c_{0}$ , $\underset{j}{lim inf} \int_{Ω} \sqrt{2 W (u_{j})} γ_{0}^{ } (\nabla u_{j}) d x ⩾ c_{0} \int_{J_{u}} γ_{0}^{ *} (ν_{u}) d H^{n - 1}$ which concludes the proof of (3.2). □

3.4. The upper estimate

We first recall some well-known results in order to construct the recovery sequence; for more details see [3]. We notice that in order to use the same construction also for the higher order part of the full energy functional we directly treat the case of $C^{2}$ -boundaries, even if for the first order energy we actually need less regularity. Let $E \subset R^{n}$ be a bounded and open set with $\partial E \cap Ω \in C^{2}$ and let $d_{E}$ be the signed distance from $\partial E \cap Ω$ , that is $d_{E} (x) : = \{\begin{matrix} d (x, \partial E) & if x \in E, \\ - d (x, \partial E) & if x \notin E . \end{matrix}$ Then, in a small tubular neighborhood of $\partial E$ it holds $d_{E} \in C^{2}$ and $| \nabla d_{E} | = 1$ . Moreover, if we let $E_{t} : = {x \in Ω : d_{E} (x) = t}$ for $t \in R$ , it turns out that $lim_{t \to 0} H^{n - 1} (E_{t}) = H^{n - 1} (\partial E \cap Ω)$ (3.3) and $Δ d_{E} = - | H_{E_{t}} | on E_{t} .$ (3.4) Let now $φ : R \to (0, 1)$ be the unique solution of the problem $\{\begin{matrix} φ^{'} (t) = \sqrt{2 W (φ (t))}, \\ lim_{t \to - \infty} φ (t) = 0, lim_{t \to + \infty} φ (t) = 1 . \end{matrix}$ (3.5) For any $ε > 0$ let $Φ_{ε} (t) : = \{\begin{matrix} φ (t) & if 0 ⩽ t < | log ε |, \\ p_{ε} (t) & if | log ε | ⩽ t ⩽ 2 | log ε |, \\ 1 & if t > 2 | log ε |, \\ - Φ_{ε} (- t) + 1 & if t < 0, \end{matrix}$ where $p_{ε}$ is a third-degree polynomial in such a way $Φ_{ε} \in C^{1, 1} (R) \cap C^{\infty} (R ∖ {\pm | log ε |, \pm 2 | log ε |}) .$ For any $t \in R$ let $φ_{ε} (t) : = Φ_{ε} (t / ε)$ . It is straightforward to verify that $φ_{ε} \to χ_{(0, + \infty)} as ε \to 0 in L^{1} (R),$ (3.6) and ${∥ φ_{ε}^{'} ∥}_{L^{\infty} (ε | log ε |, 2 ε | log ε |)} = o (ε^{2}) .$ (3.7) Let now $u_{ε} (x) : = φ_{ε} (d_{E} (x)) \forall x \in Ω .$ (3.8) Since the regularity of $d_{E}$ we obtain $u_{ε} \in H^{2} (Ω)$ for ε sufficiently small. Moreover, notice that $χ_{E} (x) = χ_{(0, + \infty)} (d_{E} (x))$ for each $x \in Ω$ ; then, by the coarea formula, since $| \nabla d_{E} | = 1$ near $\partial E \cap Ω$ , $\begin{array}{rcl} \int_{Ω} | u_{ε} (x) - χ_{E} (x) | d x & = & \int_{Ω} | φ_{ε} (d_{E} (x)) - χ_{(0, + \infty)} (d (x)) | d x \\ = & \int_{Ω} | φ_{ε} (d_{E} (x)) - χ_{(0, + \infty)} (d_{E} (x)) | | \nabla d_{E} (x) | d x \\ = & \int_{- \infty}^{+ \infty} H^{n - 1} (E_{t}) | φ_{ε} (t) - χ_{(0, + \infty)} (t) | d t . \end{array}$ Using now (3.3) and (3.6) we get, from the previous computation, $u_{ε} \to χ_{E} as ε \to 0 in L^{1} (Ω) .$ (3.9)

We are ready to start with the proof of the estimates from above. For, fix a positive infinitesimal sequence $(ε_{j})$ .

Proposition 3.4.

Let $E \subset R^{n}$ be a bounded and open set with $\partial E \cap Ω \in C^{2}$ and let $u_{j} : = u_{ε_{j}}$ , where $u_{ε_{j}}$ is given by (3.8). Then $\underset{j}{lim sup} G_{ε_{j}} (u_{j}) ⩽ c_{0} \int_{\partial E \cap Ω} γ (ν_{E}) d H^{n - 1} .$ (3.10)

Proof.

For any $j \in N$ we let $\begin{array}{rcl} A_{j} : = {x \in Ω : | d_{E} (x) | < ε_{j} | log ε_{j} |}, \\ B_{j} : = {x \in Ω : ε_{j} | log ε_{j} | < | d_{E} (x) | < 2 ε_{j} | log ε_{j} |} . \end{array}$ Since $\nabla u_{j} (x) = \frac{1}{ε_{j}} φ^{'} (\frac{d_{E} (x)}{ε_{j}}) \nabla d_{E} (x) a.e. x \in A_{j}$ (3.11) taking into account (3.5) we easily deduce that, for j sufficiently large, $\frac{ε_{j}^{2}}{2} {| \nabla u_{j} (x) |}^{2} + W (u_{j} (x)) = φ^{'} (\frac{d_{E} (x)}{ε_{j}}) \sqrt{2 W (u_{j} (x))} a.e. x \in A_{j} .$ (3.12) We first claim that $\underset{j}{lim sup} \frac{1}{ε_{j}} \int_{A_{j}} γ_{ε_{j}} (\nabla u_{j}) (\frac{ε_{j}^{2}}{2} | \nabla u_{j} |^{2} + W (u_{j})) d x ⩽ c_{0} \int_{\partial E \cap Ω} γ (ν_{E}) d H^{n - 1} .$ (3.13) Using (3.1), (3.12), and the positive one-homogeneity of $γ_{0}$ , we get $\begin{array}{rcl} \frac{1}{ε_{j}} \int_{A_{j}} γ_{ε_{j}} (\nabla u_{j} (x)) (\frac{ε_{j}^{2}}{2} {| \nabla u_{j} (x) |}^{2} + W (u_{j} (x))) d x \\ = \frac{1}{ε_{j}} \int_{A_{j}} γ_{ε_{j}} (\frac{1}{ε_{j}} φ^{'} (\frac{d_{E} (x)}{ε_{j}}) \nabla d_{E} (x)) φ^{'} (\frac{d_{E} (x)}{ε_{j}}) \sqrt{2 W (u_{j} (x))} d x \\ ⩽ \frac{1}{ε_{j}} \int_{A_{j}} γ (\nabla d_{E} (x)) φ^{'} (\frac{d_{E} (x)}{ε_{j}}) \sqrt{2 W (u_{j} (x))} d x + \frac{L r_{ε_{j}}}{2 ε_{j}} \int_{A_{j}} \sqrt{2 W (u_{j} (x))} d x . \end{array}$ Since $W (u_{j})$ is bounded, $| A_{j} | \to 0$ and $r_{ε_{j}} = O (ε_{j})$ , we have $lim_{j} \frac{r_{ε_{j}}}{ε_{j}} \int_{A_{j}} \sqrt{2 W (u_{j} (x))} d x = 0 .$ Therefore, $\begin{array}{rcl} \underset{j}{lim sup} \frac{1}{ε_{j}} \int_{A_{j}} γ_{ε_{j}} (\nabla u_{j}) (\frac{ε_{j}^{2}}{2} | \nabla u_{j} |^{2} + W (u_{j})) d x \\ ⩽ \underset{j}{lim sup} \frac{1}{ε_{j}} \int_{A_{j}} γ (\nabla d_{E} (x)) φ^{'} (\frac{d_{E} (x)}{ε_{j}}) \sqrt{2 W (u_{j} (x))} d x . \end{array}$ By coarea formula we deduce that $\begin{array}{rcl} \frac{1}{ε_{j}} \int_{A_{j}} γ (\nabla d_{E} (x)) φ^{'} (\frac{d_{E} (x)}{ε_{j}}) \sqrt{2 W (u_{j} (x))} d x \\ = \frac{1}{ε_{j}} \int_{- ε_{j} | log ε_{j} |}^{ε_{j} | log ε_{j} |} \int_{E_{t}} γ (\nabla d_{E} (x)) φ^{'} (\frac{d_{E} (x)}{ε_{j}}) \sqrt{2 W (φ^{'} (\frac{d_{E} (x)}{ε_{j}}))} d H^{n - 1} (x) d t \\ = \frac{1}{ε_{j}} \int_{- ε_{j} | log ε_{j} |}^{ε_{j} | log ε_{j} |} φ^{'} (\frac{t}{ε_{j}}) \sqrt{2 W (φ (\frac{t}{ε_{j}}))} \int_{E_{t}} γ (\nabla d_{E} (x)) d H^{n - 1} (x) d t \\ = \int_{- | log ε_{j} |}^{| log ε_{j} |} φ^{'} (s) \sqrt{2 W (φ (s))} \int_{E_{ε_{j} s}} γ (\nabla d_{E} (x)) d H^{n - 1} (x) d s . \end{array}$ For each t with $| t |$ small let $F^{t} \subseteq R^{n}$ be such that $\partial F^{t} \cap Ω = E_{t}$ . For any $s \in R$ we easily get $D χ_{F^{ε_{j} s}} ⇀^{*} H^{n - 1} ⌞ (\partial E \cap Ω)$ as $j \to + \infty$ ; moreover, since (3.3) holds we also get $| D χ_{F^{ε_{j} s}} | (Ω) \to H^{n - 1} (\partial E \cap Ω)$ as $j \to + \infty$ . For each $x \in E_{ε_{j} s}$ we have $\frac{d D χ_{F^{ε_{j} s}}}{d | D χ_{F^{ε_{j} s}} |} (x) = \nabla d_{E} (x)$ and therefore $\int_{E_{ε_{j} s}} γ (\nabla d_{E} (x)) d H^{n - 1} (x) = \int_{Ω} γ (\frac{d D χ_{F^{ε_{j} s}}}{d | D χ_{F^{ε_{j} s}} |} (x)) d | D χ_{F^{ε_{j} s}} | (x) .$ Since γ is continuous and bounded and we have convergence of the masses, applying Reshetnyak Continuity theorem we can say that $lim_{j \to + \infty} \int_{Ω} γ (\frac{d D χ_{F^{ε_{j} s}}}{d | D χ_{F^{ε_{j} s}} |} (x)) d | D χ_{F^{ε_{j} s}} | (x) = \int_{\partial E \cap Ω} γ (ν_{E}) d H^{n - 1} .$ Finally, observing that $\begin{array}{rcl} \int_{- | log ε_{j} |}^{| log ε_{j} |} φ^{'} (s) \sqrt{2 W (φ (s))} \int_{E_{ε_{j} s}} γ (\nabla d_{E} (x)) d H^{n - 1} (x) d s \\ ⩽ c \int_{- \infty}^{+ \infty} φ^{'} (s) \sqrt{2 W (φ (s))} H^{n - 1} (E_{ε_{j} s}) d s, \end{array}$ using the Dominated Convergence theorem we deduce that $\begin{array}{rcl} lim_{j} \frac{1}{ε_{j}} \int_{A_{j}} γ (\nabla d_{E} (x)) φ^{'} (\frac{d_{E} (x)}{ε_{j}}) \sqrt{2 W (u_{j} (x))} d x \\ = \int_{- \infty}^{+ \infty} φ^{'} (s) \sqrt{2 W (φ (s))} d s \int_{\partial E \cap Ω} γ (ν_{E}) d H^{n - 1} . \end{array}$ The proof of (3.13) is now complete: it is sufficient to use (3.5) and the very definition of $c_{0}$ . In order to conclude it remains to prove that $\underset{j}{lim sup} \frac{1}{ε_{j}} \int_{B_{j}} γ_{ε_{j}} (\nabla u_{j}) (\frac{ε_{j}^{2}}{2} | \nabla u_{j} |^{2} + W (u_{j})) d x = 0$ (3.14) since $\int_{Ω ∖ (A_{j} \cup B_{j})} γ_{ε_{j}} (\nabla u_{j}) (\frac{ε_{j}^{2}}{2} | \nabla u_{j} |^{2} + W (u_{j})) d x = 0$ for any j. Using the very definition of $u_{j}$ and (3.7), we can say that $γ_{ε_{j}} (\nabla u_{j})$ is bounded on $B_{j}$ , and thus we find that $\begin{array}{rcl} \frac{1}{ε_{j}} \int_{B_{j}} γ_{ε_{j}} (\nabla u_{j}) (\frac{ε_{j}^{2}}{2} | \nabla u_{j} |^{2} + W (u_{j})) d x \\ ⩽ \frac{c}{ε_{j}} \int_{B_{j}} (\frac{ε_{j}^{2}}{2} | \nabla u_{j} |^{2} + W (u_{j})) d x \\ = \frac{c}{ε_{j}} \int_{B_{j}} (\frac{1}{2} {| φ_{ε_{j}}^{'} (\frac{d_{E} (x)}{ε_{j}}) |}^{2} + W (φ_{ε_{j}} (\frac{d_{E} (x)}{ε_{j}}))) d x . \end{array}$ Using again the coarea formula, it follows that $\begin{array}{rcl} \frac{1}{ε_{j}} \int_{B_{j}} (\frac{1}{2} {| φ_{ε_{j}}^{'} (\frac{d_{E} (x)}{ε_{j}}) |}^{2} + W (φ_{ε_{j}} (\frac{d_{E} (x)}{ε_{j}}))) d x \\ = \int_{| log ε_{j} |}^{2 | log ε_{j} |} (\frac{1}{2} {| φ_{ε_{j}}^{'} (s) |}^{2} + W (φ_{ε_{j}} (s))) H^{n - 1} (E_{ε_{j} s}) d s . \end{array}$ Combining (3.3) with (3.7) we obtain $\underset{j}{lim sup} \int_{| log ε_{j} |}^{2 | log ε_{j} |} {| φ_{ε_{j}}^{'} (s) |}^{2} H^{n - 1} (E_{ε_{j} s}) d s = 0$ while by the Dominated Convergence theorem we get $\underset{j}{lim sup} \int_{| log ε_{j} |}^{2 | log ε_{j} |} W (φ_{ε_{j}} (s)) H^{n - 1} (E_{ε_{j} s}) d s = 0$ since (3.3) and (3.6) hold true and, using the very definition of W, $W (φ_{ε_{j}} (s)) ⩽ \frac{| 1 - φ_{ε_{j}} (s) |}{2} \forall s ⩾ 0 .$ Combining all these facts we deduce (3.14). □

We are now in position to complete the estimate from above by means of a relaxation argument. In order to do this, we recall that, by standard Γ-convergence theory, we have to prove that $G^{″} (u) ⩽ G (u)$ everywhere, where $G^{″}$ is the so called Γ-lim sup given by $G^{″} (u) : = inf {\underset{j}{lim sup} G_{ε_{j}} (u_{j}) : u_{j} \to u in L^{1} (Ω)} .$ We also recall that $G^{″}$ turns out to be always a lower semicontinuous functional. Proposition 3.5.

For each $u \in L^{1} (Ω)$ we have $G^{″} (u) ⩽ G (u)$ .

Proof.

Of course, it is sufficient to consider the case $u \in BV (Ω; {0, 1})$ . Formula (3.10) says that for any $E \subset R^{n}$ bounded, open and with $\partial E \cap Ω \in C^{2}$ , it holds $G^{″} (χ_{E}) ⩽ c_{0} \int_{\partial E \cap Ω} γ (ν_{E}) d H^{n - 1} .$ Fix $u \in BV (Ω; {0, 1})$ ; in particular, $u = χ_{E}$ , where E is a set with finite perimeter in Ω. Let $(E_{h})$ be a sequence of bounded, open sets with $\partial E_{h} \cap Ω \in C^{2}$ for any $h \in N$ , such that $χ_{E_{h}} \to u$ in $L^{1} (Ω)$ and $lim_{h} | D χ_{E_{h}} | (Ω) = | D u | (Ω);$ such a sequence always exists, see for instance Theorem 3.42 in [2]. Let $μ_{h} : = D χ_{E_{h}}$ . Using again Reshetnyak Continuity theorem and taking into account that $G^{″}$ is lower semicontinuous with respect to the $L^{1}$ -convergence, we get $\begin{array}{rcl} G^{″} (u) & ⩽ & c_{0} \underset{h \to + \infty}{lim inf} \int_{\partial E_{h} \cap Ω} γ (ν_{E_{h}}) d H^{n - 1} = c_{0} \underset{h \to + \infty}{lim inf} \int_{Ω} γ (\frac{d μ_{h}}{d | μ_{h} |}) d | μ_{h} | \\ = & c_{0} \int_{\partial E \cap Ω} γ (ν_{E}) d H^{n - 1} = c_{0} \int_{J_{u}} γ (ν_{u}) d H^{n - 1} . \end{array}$ Therefore, we have that $G^{″} (u) ⩽ c_{0} \int_{J_{u}} γ (ν_{u}) d H^{n - 1}$ holds true for each $u \in BV (Ω; {0, 1})$ . Let $J : L^{1} (Ω) \to [0, + \infty]$ be given by $J (u) : = \{\begin{matrix} \int_{J_{u}} γ (ν_{u}) d H^{n - 1} & if u \in BV (Ω; {0, 1}), \\ + \infty & otherwise . \end{matrix}$ Since $G^{″}$ is lower semicontinuous with respect to the $L^{1}$ -convergence, it holds $G^{″} (u) ⩽ \bar{J} (u)$ where $\bar{J}$ is the $L^{1}$ -relaxed functional of J. It is well known that $\bar{J}$ takes the form $\bar{J} (u) = \{\begin{matrix} \int_{J_{u}} E γ (ν_{u}) d H^{n - 1} & if u \in BV (Ω; {0, 1}), \\ + \infty & otherwise, \end{matrix}$ where $E γ : S^{n - 1} \to [0, + \infty)$ is the $BV$ -elliptic envelope of $γ_{|_{S^{n - 1}}}$ (see [1] for details), that is the bigger $BV$ -elliptic function less than $γ_{|_{S^{n - 1}}}$ . In order to conclude the proof it is sufficient to show that $E γ ⩽ γ_{0}^{* *}$ . First of all, since $E γ ⩽ γ$ , we also deduce that ${(E γ)}_{0} ⩽ γ_{0}$ , where we have denoted by ${(E γ)}_{0} : R^{n} \to [0, + \infty)$ the positively one-homogeneous extension of $E γ$ . But since ${(E γ)}_{0} = E γ$ on $S^{n - 1}$ the functional $u \mapsto \{\begin{matrix} \int_{J_{u}} {(E γ)}_{0} (ν_{u}) d H^{n - 1} & if u \in BV (Ω; {0, 1}), \\ + \infty & otherwise \end{matrix}$ remains lower semicontinuous, and then, taking into account, for instance, Theorem 5.11 in [2], it descends that ${(E γ)}_{0}$ is convex, so that ${(E γ)}_{0} ⩽ γ_{0}^{* *}$ , which gives $E γ ⩽ γ_{0}^{* *}$ , and this yields the conclusion. □

4. Some remarks about the full energy

In this section we consider the full energy $F_{ε} : L^{1} (Ω) \to [0, + \infty]$ given by $F_{ε} : = G_{ε} + H_{ε}$ , where $H_{ε} (u) : = \{\begin{matrix} \frac{β}{2 ε} \int_{Ω} {(- ε Δ u + \frac{1}{ε} W^{'} (u))}^{2} d x & if u \in H^{2} (Ω), \\ + \infty & otherwise in L^{1} (Ω) \end{matrix}$ and $β > 0$ is a fixed constant. Following Röger and Schätzle (see [11]) it is possible to prove, by means of varifolds approach, that when $n = 2, 3$ and $E \subset R^{n}$ is bounded and open with $\partial E \cap Ω \in C^{2}$ for any sequence $(u_{j})$ in $H^{2} (Ω)$ with $u_{j} \to χ_{E}$ strongly in $L^{1} (Ω)$ and with $sup_{j} \frac{1}{ε_{j}} \int_{Ω} (\frac{ε_{j}^{2}}{2} | \nabla u_{j} |^{2} + W (u_{j})) d x < + \infty$ it holds $\underset{j}{lim inf} H_{ε_{j}} (u_{j}) ⩾ \frac{c_{0} β}{2} \int_{\partial E \cap Ω} H_{\partial E \cap Ω}^{2} d H^{n - 1},$ where we remember that $H_{\partial E \cap Ω}$ denotes the mean curvature of $\partial E \cap Ω$ . Therefore, by the subadditivity of the lim inf operator we deduce that $\underset{j}{lim inf} F_{ε_{j}} (u_{j}) ⩾ c_{0} \int_{\partial E \cap Ω} γ_{0}^{* *} (ν_{E}) d H^{n - 1} + \frac{c_{0} β}{2} \int_{\partial E \cap Ω} H_{\partial E \cap Ω}^{2} d H^{n - 1} .$ (4.1) Concerning the Γ-lim sup estimate we notice first that for the same sequence $u_{j}$ used for the upper estimate for $G_{ε}$ we get, always when $n = 2, 3$ and $E \subset R^{n}$ is bounded and open with $\partial E \cap Ω \in C^{2}$ , $\underset{j}{lim sup} H_{ε_{j}} (u_{j}) ⩽ \frac{c_{0} β}{2} \int_{\partial E \cap Ω} H_{\partial E \cap Ω}^{2} d H^{n - 1}$ and this fact has been proven in [3]. Combining (3.10) with the superadditivity of the lim sup operator we obtain $\underset{j}{lim sup} F_{ε_{j}} (u_{j}) ⩽ c_{0} \int_{\partial E \cap Ω} γ (ν_{E}) d H^{n - 1} + \frac{c_{0} β}{2} \int_{\partial E \cap Ω} H_{\partial E \cap Ω}^{2} d H^{n - 1} .$ (4.2) Taking into account (4.1) with (4.2) we deduce the following theorem.

Theorem 4.1.

Assume $n = 2, 3$ and $γ_{0}$ convex. Then, for each $E \subset R^{n}$ bounded, open, with $\partial E \cap Ω \in C^{2}$ we have $\underset{ε \to 0}{Γ - lim} F_{ε} (χ_{E}) = c_{0} \int_{\partial E \cap Ω} γ (ν_{E}) d H^{n - 1} + \frac{c_{0} β}{2} \int_{\partial E \cap Ω} H_{\partial E \cap Ω}^{2} d H^{n - 1}$ with respect to the strong $L^{1}$ -convergence.

Footnotes

Acknowledgements

Part of this work has been done during the permanence of the author at the Carnegie Mellon University in Pittsburgh (PA, USA), in collaboration with Irene Fonseca and Giovanni Leoni: the author thanks Irene and Giovanni for a lot of suggestions and remarks. Moreover, this work was partially supported by a project GNAMPA 2010: Problemi variazionali in micromagnetismo.

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