Γ -type estimates for the one-dimensional Allen

Abstract

In this paper we prove an asymptotic estimate, up to the second-order included, on the behaviour of the one-dimensional Allen–Cahn’s action functionals, around a periodic function with bounded variation and taking values in ${\pm 1}$ . The leading term of this estimate justifies and confirms, from a variational point of view, the results of Fusco–Hale [Dyn. Diff. Equation 1 (1989), 75–94] and Carr–Pego [Comm. Pure Appl. Math. 42 (1989), 523–576] on the exponentially slow motion of metastable patterns coexisting at the transition temperature.

Keywords

singular perturbations Γ-expansion Allen–Cahn’s action

1. Introduction

In this paper we are interested in the asymptotic behaviour as $ε \to 0^{+}$ of the one-dimensional Allen–Cahn’s action functionals $F_{ε} (u) : = \int_{T} (\frac{ε}{2} {(u^{'})}^{2} + \frac{W (u)}{ε}) d x,$ where $T$ is the one-dimensional unit torus, W is a smooth double well potential with zeroes at $\pm 1$ , and $u : T \to R$ . These functionals arise in several models of phase transitions in materials science, see for instance [4,8,11–13] and references therein. In particular, two phases $u = \pm 1$ , coexisting at the transition temperature, exhibit metastable patterns which slowly evolve according to the $L^{2}$ -gradient flow of $F_{ε}$ , $u_{t} = ε^{2} u_{x x} - W^{'} (u),$ (1.1) where a time rescaling has been performed. Equation (1.1) is perhaps the simplest partial differential equation modelling nonlinear relaxation to equilibrium in the presence of competing stable states. In [8,11] the authors showed that, as $ε \to 0^{+}$ , a solution u of (1.1) is locally equal to $\pm 1$ and the transition points evolve, exponentially slowly, in accordance to a specific system of ODEs (see [11, Eq. (3.11)] and [8, Eq. (1.2)]). The exponential speed is dictated by the qualitative properties of W, in particular by its nondegeneracy at $\pm 1$ .

In this paper we aim to provide a variational counterpart of the dynamical results of [8,11], recovering an analogous ODEs system obtained as a by-product of the behaviour, at the leading order, of the action functionals $F_{ε}$ for $ε ≪ 1$ , around piecewise constant functions u with values in ${\pm 1}$ , which correspond to the metastable patterns in the two-phase model described above. It is well known [10,14] that the sequence $(F_{ε})$ is equicoercive in $L^{1} (T)$ and Γ- $L^{1} (T)$ -converges, as $ε \to 0^{+}$ , to the functional $F_{0} : L^{1} (T) \to [0, + \infty]$ defined as $F_{0} (u) : = \{\begin{matrix} N (u) σ & if u \in BV (T; {\pm 1}), \\ + \infty & otherwise, \end{matrix}$ (1.2) where $N (u)$ is the number of jump points of u, and where $σ : = inf {\int_{R} (\frac{1}{2} {(v^{'})}^{2} + W (v)) d y : v \in H_{loc}^{1} (R), v (0) = 0, lim_{y \to \pm \infty} v (y) = \pm 1}$ (1.3) is sometimes called surface tension.

The main results of this paper are the following asymptotic estimates. Firstly (Theorem 6.1) we prove that $\begin{array}{rclr} F_{ε} (v_{ε}) & ⩾ & N (v) σ - α_{+} κ_{+}^{2} \sum_{k = 1}^{N (v)} e^{- α_{+} (d_{k}^{ε} / ε)} - α_{-} κ_{-}^{2} \sum_{k = 1}^{N (v)} e^{- α_{-} (d_{k}^{ε} / ε)} \\ + o (\sum_{k = 1}^{N (v)} e^{- α_{+} (d_{k}^{ε} / ε)}) + o (\sum_{k = 1}^{N (v)} e^{- α_{-} (d_{k}^{ε} / ε)}) & (1.4) \end{array}$ as $ε \to 0^{+}$ , where $(v_{ε})$ is any sequence converging to $v \in BV (T; {\pm 1})$ in $L^{1} (T)$ , $α_{\pm}$ , $κ_{\pm}$ are constants1

$κ_{\pm}$ are defined in (3.7), (3.8).

depending on W, in particular

α_{\pm} : = \sqrt{W^{″} (\pm 1)}

, and

d_{k}^{ε}

is the distance between the kth and the

(k + 1)

th transition of

v^{ε}

(see (5.9) and (6.2)). Notice that the terms appearing on the right-hand side of (1.4) scale differently in ε, as soon as the limits

d_{k} (v) : = {lim}_{ε} d_{k}^{ε}

are different for different k’s, and in particular we cannot substitute the approximate distance

d_{k}^{ε}

with the distance

d_{k} (v) = x_{k + 1} (v) - x_{k} (v)

between the consecutive kth and

(k + 1)

th jump point of the limit function v. Estimate (1.4) is sharp, in the sense that for any

v \in BV (T; {\pm 1})

there exists a sequence

(v_{ε})

such that the equality holds in (1.4) (Theorem 6.5).

Secondly (Theorem 8.3) we show that if W is a parabola near $\pm 1$ then we can improve (1.4), obtaining the (sharp) estimate $\begin{array}{rclr} F_{ε} (v_{ε}) & ⩾ & N (v) σ - α_{+} κ_{+}^{2} \sum_{k = 1}^{N (v)} e^{- α_{+} (d_{k}^{ε} / ε)} - α_{-} κ_{-}^{2} \sum_{k = 1}^{N (v)} e^{- α_{-} (d_{k}^{ε} / ε)} \\ + o (\sum_{k = 1}^{N (v)} e^{- (3 α_{+} / 2) (d_{k}^{ε} / ε)}) + o (\sum_{k = 1}^{N (v)} e^{- (3 α_{-} / 2) (d_{k}^{ε} / ε)}) . & (1.5) \end{array}$ Observe that (1.5) provides a sort of second-order asymptotic expansion (with vanishing second-order term) of $F_{ε}$ around functions $v \in BV (T; {\pm 1})$ , which is reminiscent of a Γ-expansion of $F_{ε}$ in the sense of [1,2,5,6]. However, our results cannot be straightforwardly framed in that setting: for instance, we do not restrict ourselves to expand around a global minimizer of $F_{0}$ (which would be the constant state $u \equiv 1$ or $u \equiv - 1$ ), but we need to work around an $L^{1} (T)$ -local minimizer, more specifically around a periodic bounded variation function taking values in $\pm 1$ . In Section 7 we associate to our first order estimate (1.4) a Γ-limit, which turns out to be unbounded from below.

Now, let us introduce the functional $G_{ε} : BV (T; {\pm 1}) \to R$ as $G_{ε} (v) : = N (v) σ - \sum_{k = 1}^{N (v)} (α_{+} κ_{+}^{2} e^{- α_{+} (d_{k} (v) / ε)} + α_{-} κ_{-}^{2} e^{- α_{-} (d_{k} (v) / ε)}) .$ (1.6) Notice that the right-hand side of (1.4) coincides, at the leading order, with $G_{ε} (v)$ where however, the approximate distance $d_{k}^{ε}$ is replaced by the distance $d_{k} (v)$ .

The functional $G_{ε}$ may be considered as a function of $N (v)$ variables, that is $G_{ε} (v) = G_{ε} (x_{1} (v), x_{2} (v), \dots, x_{N (v)} (v)),$ (1.7) and the gradient flow of $G_{ε}$ is given by the system of ODEs $\begin{array}{rclr} {\dot{x}}_{j} (v) & = & - \frac{\partial G_{ε} (v)}{\partial x_{j} (v)} \\ = & \frac{α_{+}^{2} κ_{+}^{2}}{ε} (e^{- α_{+} ((x_{j + 1} (v) - x_{j} (v)) / ε)} - e^{- α_{+} ((x_{j} (v) - x_{j - 1} (v)) / ε)}) \\ + \frac{α_{-}^{2} κ_{-}^{2}}{ε} (e^{- α_{-} ((x_{j + 1} (v) - x_{j} (v)) / ε)} - e^{- α_{-} ((x_{j} (v) - x_{j - 1} (v)) / ε)}) & (1.8) \end{array}$ for all $j = 1, \dots, N (v)$ , where ${\dot{x}}_{j}$ stands for the derivative of $x_{j}$ with respect to t. Notice from (1.8) that the jump point $x_{j} (v)$ interacts only with $x_{j - 1} (v)$ and $x_{j + 1} (v)$ .

An interesting observation is that system (1.8) coincides, up to a multiplicative constant,2

The presence of a multiplicative constant is not surprising, for instance, a similar phenomenon happens in mean curvature flow when approximated with the parabolic Allen–Cahn’s equation. Such constants can be normalized to one by a time scaling.

with the evolution equations obtained in [11, Eq. (3.11)], [8, Eq. (1.2)], thus showing the consistency of (1.4) with the behaviour of (1.1) as

ε \to 0^{+}

. This is in accordance with the general principle outlined in [15,16], where the authors relate the gradient flow of the Γ-limit of a sequence of functionals with the limit of the gradient flows.

We observe that, not surprisingly, only the terms of order $e^{- α_{\pm} (d_{k}^{ε} / ε)}$ are relevant for the evolution law of the jump points $x_{j} (v)$ .

We conclude this introduction by mentioning that the results of [11] and [8] have been generalized to a vector setting (in the target space) in the paper [3]; generalizing estimate (1.4) to this more general situation seems, however, not easy.

The content of the paper is the following. In Section 2 we set the notation. In Section 3 we introduce the constants $c_{\pm}$ and hence the constants $κ_{\pm}$ appearing in (1.4), (1.5) and (1.8). In Section 4 we introduce various functions, which are useful to prove the main results. The expansions of those functions are computed in Section 4.1. Two lemmas, based on variational arguments, and necessary to the main results are next proven in Section 5. We prove estimate (1.4) and its sharpness in Theorems 6.1 and 6.5. In Section 7 we find a related first order Γ-limit, under the additional assumption that W is even. This is the only place of the paper where we make such an assumption. Eventually, in Section 8 we prove (1.5): as mentioned above, we are able to show this estimate supposing that W is a parabola near its minimum point, and this makes easy to treat the various singular integrals involved (in particular, the derivative of the function $D_{+}$ , defined in (4.5), evaluated at the point $s = 1$ .

2. Notation

The assumptions on the double well potential W are the following:

$W : R \to [0, + \infty)$ and $W \in C^{\infty} (R)$ ;

$W^{- 1} (0) = {\pm 1}$ ;

$W^{″} (\pm 1) > 0$ . We set $α_{+} : = \sqrt{W^{″} (1)}, α_{-} : = \sqrt{W^{″} (- 1)} .$ (2.1)

Notice that we do not suppose that W is even. We define

β_{\pm} : = W^{‴} (\pm 1) .

A Taylor expansion around

s = 1

gives

2 W (s) = α_{+}^{2} {(1 - s)}^{2} [1 + \frac{β_{+}}{3 α_{+}^{2}} (s - 1)] + o ({(1 - s)}^{3}) as s \to 1^{-},

(2.2) and similarly in a right neighborhood of

- 1

with

α_{-}

replacing

α_{+}

and

β_{-}

replacing

β_{+}

From (2.2) it follows $\frac{1}{\sqrt{2 W (s)}} = \frac{1}{α_{+} (1 - s)} (1 + \frac{β_{+}}{6 α_{+}^{2}} (1 - s) + o (1 - s))$ as $s \to 1^{-}$ , so that $\frac{1}{\sqrt{2 W (s)}} - \frac{1}{α_{+} (1 - s)} = \frac{β_{+}}{6 α_{+}^{3}} + ρ (s),$ (2.3) where the reminder $ρ : [0, 1] \to R is continuous, lim_{s \to 1^{-}} ρ (s) = 0 .$ We define $ϕ (η) : = \int_{- 1}^{η} \sqrt{2 W (s)} d s, η \in [- 1, 1] .$ (2.4) It is known that σ in (1.3) satisfies $σ = ϕ (1) = σ_{-} + σ_{+},$ where we have set $σ_{-} : = \int_{- 1}^{0} \sqrt{2 W (s)} d s, σ_{+} : = \int_{0}^{1} \sqrt{2 W (s)} d s .$ Remark 2.1.

Our assumptions on W ensure that for $η \in (0, 1)$ (resp. $η \in (- 1, 0)$ ) sufficiently close to 1 (resp. to $- 1$ ) we have $W (η) < W (s), s \in (0, η) (resp. s \in (η, 0)),$ and $W^{'} (η) < 0$ (resp. $W^{'} (η) > 0$ ).

2.1. Periodic

BV

functions

Let $T$ be the one-dimensional unit torus. We denote by $BV (T; {\pm 1})$ the space of functions of bounded variation in $T$ taking values $\pm 1$ . For a function $u \in BV (T; {\pm 1})$ with nonempty jump set $S (u) \subset T$ , we write $S (u) = {x_{1} (u), \dots, x_{N (u)} (u)}$ , where $N (u) \in (0, + \infty)$ is the number of the jump points of u, and $x_{1} (u) < x_{2} (u) < \dots < x_{N (u)} (u) < x_{N (u) + 1} (u) : = x_{1} (u) .$ (2.5) We let $N_{+} (u)$ be the number of increasing jumps from $- 1$ to 1 (resp. $N_{-} (u)$ be the number of decreasing jumps from 1 to $- 1$ ). Due to the periodicity of functions in $BV (T; {\pm 1})$ , $N (u)$ is even (or zero) and $N_{+} (u) = N_{-} (u)$ . If $S (u) = \emptyset$ we set $N (u) = N_{+} (u) = N_{-} (u) = 0$ . Definition 2.2.

Let $u \in BV (T; {\pm 1})$ be nonconstant. We define $\{\begin{matrix} d_{k} (u) : = x_{k + 1} (u) - x_{k} (u), k = 1, \dots, N (u) - 1, \\ d_{N (u)} : = 1 - (x_{N (u)} - x_{1} (u)) = d_{0} (u), \end{matrix}$ (2.6) and $\begin{array}{rcl} I_{+} (u) : = {k \in {1, \dots, N (u)} : u jumps from - 1 to 1 at x_{k} (u)}, \\ I_{-} (u) : = {1, \dots, N (u)} ∖ I_{+} (u) . \end{array}$

2.2. The functionals

F_{ε}

and the minimizer γ

For any $ε \in (0, 1)$ let $F_{ε} : L^{1} (T) \to [0, + \infty]$ be defined by $F_{ε} (u) : = \{\begin{matrix} \int_{T} (\frac{ε}{2} {(u^{'})}^{2} + \frac{W (u)}{ε}) d x & if u \in H^{1} (T) and W (u) \in L^{1} (T), \\ + \infty & otherwise . \end{matrix}$ (2.7) When I is a measurable subset of $T$ , we denote by $F_{ε} (\cdot, I)$ the localization of $F_{ε} (\cdot)$ on I (obtained by replacing $T$ with I in (2.7)) and we set $F_{ε} (\cdot, T) = F_{ε} (\cdot)$ .

If $J \subset R$ is a bounded interval and $v \in H^{1} (J)$ , we set $F (v, J) : = \int_{J} (\frac{1}{2} {(v^{'})}^{2} + W (v)) d y,$ (2.8) and for $v \in H_{loc}^{1} (R)$ , we let $F (v) : = \int_{R} (\frac{1}{2} {(v^{'})}^{2} + W (v)) d y$ .

It is well known that the infimum in (1.3) is a minimum and is attained by the function $γ \in C^{\infty} (R)$ solving $\{\begin{matrix} γ^{'} = \sqrt{2 W (γ)} in R, \\ γ (0) = 0 . \end{matrix}$ (2.9)

3. The functions

B_{\pm}

, the constants

c_{\pm}

and

κ_{\pm}

We let $B_{+} (η) : = \int_{0}^{η} \frac{1}{\sqrt{2 W (s)}} d s, η \in (0, 1) .$ (3.1) Note that $B_{+} (η) = \frac{1}{α_{+}} log (\frac{1}{1 - η}) + o (1) log (1 - η) as η \to 1^{-} .$ (3.2) Indeed, from (2.3) and de l’Hôpital theorem, we deduce $lim_{η \to 1^{-}} \frac{B_{+} (η)}{- log (1 - η)} = lim_{η \to 1^{-}} \frac{1 - η}{\sqrt{2 W (η)}} = \frac{1}{α_{+}},$ and (3.2) follows.

Lemma 3.1 (Expansion of

B_{+}

Let $B_{+}$ be the function defined in (3.1). Then $\exists lim_{η \to 1^{-}} (B_{+} (η) - \frac{1}{α_{+}} log (\frac{1}{1 - η})) = : c_{+} \in R .$ (3.3) Hence $B_{+} (η) = \frac{1}{α_{+}} log (\frac{1}{1 - η}) + c_{+} + o (1) as η \to 1^{-} .$ (3.4)

Proof.

We have $B_{+} (η) - \frac{1}{α_{+}} log (\frac{1}{1 - η}) = \int_{0}^{η} (\frac{1}{\sqrt{2 W (s)}} - \frac{1}{α_{+} (1 - s)}) d s, η \in (0, 1) .$ (3.5) Coupling (2.3) with (3.5) we get $B_{+} (η) - \frac{1}{α_{+}} log (\frac{1}{1 - η}) = \frac{β_{+}}{6 α_{+}^{3}} η + \int_{0}^{η} ρ (s) d s, η \in (0, 1) .$ (3.6) Then formula (3.3) follows, and $\begin{array}{rclr} c_{+} & = & lim_{η \to 1^{-}} \int_{0}^{η} (\frac{1}{\sqrt{2 W (s)}} - \frac{1}{α_{+} (1 - s)}) d s = \int_{0}^{1} (\frac{1}{\sqrt{2 W (s)}} - \frac{1}{α_{+} (1 - s)}) d s \\ = & \frac{β_{+}}{6 α_{+}^{3}} + \int_{0}^{1} ρ (s) d s . □ & (3.7) \end{array}$

The minimizer γ tends to its asymptotic values with an exponential rate given by $α_{\pm}$ . For convenience of the reader and for future reference (see the proof of Theorem 6.1) we give the proof of the following result (see for instance [8]).

Corollary 3.2 (Asymptotic behaviour of γ).

There exist the limits $lim_{y \to + \infty} \frac{1 - γ (y)}{e^{- α_{+} y}} = : κ_{+} \in (0, + \infty), lim_{y \to - \infty} \frac{1 + γ (y)}{e^{α_{-} y}} = : κ_{-} \in (0, + \infty) .$ (3.8)

Proof.

We consider the case $y > 0$ , the case $y < 0$ being similar. From (2.9) it follows $\int_{0}^{y} \frac{γ^{'}}{\sqrt{2 W (γ)}} d z = B_{+} (γ (y)) = y, y > 0 .$ (3.9) Therefore, using (3.4) we find $y = - \frac{1}{α_{+}} log (1 - γ (y)) + c_{+} + o (1)$ as $y \to + \infty$ . This implies the assertion in (3.8) with $κ_{+} = e^{α_{+} c_{+}} . □$ (3.10)

We set $B_{-} (η) : = \int_{η}^{0} \frac{1}{\sqrt{2 W (s)}} d s$ for $η \in (- 1, 0)$ , and $c_{-} : = lim_{η \to 1^{-}} \int_{0}^{η} (\frac{1}{\sqrt{2 W (- s)}} - \frac{1}{α_{-} (1 + s)}) d s, κ_{-} : = e^{α_{-} c_{-}} .$

4. The functions

Q_{\pm}

A_{\pm}

D_{\pm}

L_{\pm}

From Remark 2.1 we have that the function $Q_{+} (η) : = \int_{0}^{η} \frac{1}{\sqrt{2 W (s) - 2 W (η)}} d s, η \in (0, 1) close enough to 1$ (4.1) is well defined (one checks that $\frac{1}{\sqrt{2 W (\cdot) - 2 W (η)}} \in L^{1} (0, η)$ ).

We let $A_{+} (η) : = \frac{1}{α_{+}} \int_{0}^{η} \frac{1}{\sqrt{{(1 - s)}^{2} - {(1 - η)}^{2}}} d s, η \in (0, 1) .$ (4.2) Setting $1 - η = ξ$ , changing variable with $1 - s = t$ and then $t / ξ = x$ we get $A_{+} (η) = \frac{1}{α_{+}} \int_{1}^{1 / (1 - η)} \frac{1}{\sqrt{x^{2} - 1}} d x$ . With a direct integration we have $A_{+} (η) = \frac{1}{α_{+}} log (\frac{1 - η}{1 - \sqrt{1 - {(1 - η)}^{2}}}), η \in (0, 1) .$ (4.3) Hence3

Notice that if we put $η = 1$ inside the integrand of (4.2) we get $\frac{1}{α_{+}} \int_{0}^{η} \frac{1}{1 - s} d s$ , which is not equal to the leading term on the right-hand side of (4.4).

A_{+} (η) = \frac{1}{α_{+}} log (\frac{2}{1 - η}) + o (1) as η \to 1^{-} .

(4.4) For

η \in (0, 1)

sufficiently close to 1, we also consider the difference

D_{+} (η) : = Q_{+} (η) - A_{+} (η) .

(4.5) Finally

L_{+} (η) : = \int_{0}^{η} \sqrt{2 W (s) - 2 W (η)} d s, η \in (0, 1] close enough to 1 .

(4.6) Notice that

lim_{η \to 1^{-}} L_{+} (η) = L_{+} (1) = σ_{+}

and

L_{+}^{'} (η) = - W^{'} (η) Q_{+} (η), η \in (0, 1) close enough to 1 .

(4.7)

We set $Q_{-} (η) : = \int_{η}^{0} \frac{1}{\sqrt{2 W (s) - 2 W (η)}} d s$ , $L_{-} (η) : = \int_{η}^{0} \sqrt{2 W (s) - 2 W (η)} d s$ , and $D_{-} (η) : = Q_{-} (η) - A_{-} (η)$ for $η \in (- 1, 0)$ sufficiently close to $- 1$ , where $A_{-} (η) : = \frac{1}{α_{-}} \int_{η}^{0} \frac{1}{\sqrt{{(1 - s)}^{2} - {(1 - η)}^{2}}} d s$ for $η \in (- 1, 0)$ .

4.1. Expansions of

D_{+}

Q_{+}

and

L_{+}

We shall need the following result.

Lemma 4.1 (Expansion of $D_{+}$ ).

Let $c_{+}$ be as in (3.3). Then $lim_{η \to 1^{-}} D_{+} (η) = c_{+} = D_{+} (1) .$ (4.8)

Proof.

Recalling (2.2), write $f (x) : = 2 W (x) - α_{+}^{2} {(1 - x)}^{2} - \frac{β_{+}}{3} {(x - 1)}^{3}, x \in R .$ For $s < η < 1$ we write $f (s) - f (η) = f^{'} (ξ) (s - η)$ for a suitable $ξ \in (s, η)$ : for s sufficiently close to 1 we deduce $\begin{array}{rclr} 2 W (s) - 2 W (η) \\ = α_{+}^{2} ({(1 - s)}^{2} - {(1 - η)}^{2}) + \frac{β_{+}}{3} ({(s - 1)}^{3} - {(η - 1)}^{3}) \\ + (2 W^{'} (ξ) + 2 α_{+}^{2} (1 - ξ) - β_{+} {(1 - ξ)}^{2}) (s - η) \\ = α_{+}^{2} ({(1 - s)}^{2} - {(1 - η)}^{2}) (1 + \frac{β_{+}}{3 α_{+}^{2}} ψ (s, η) + R (s, η)), & (4.9) \end{array}$ where $\{\begin{matrix} ψ (s, η) = \frac{{(s - 1)}^{3} - {(η - 1)}^{3}}{{(1 - s)}^{2} - {(1 - η)}^{2}} = - \frac{{(1 - s)}^{2} + (1 - s) (1 - η) + {(1 - η)}^{2}}{(1 - s) + (1 - η)}, \\ R (s, η) = \frac{(2 W^{'} (ξ) + 2 α_{+}^{2} (1 - ξ) - β_{+} {(1 - ξ)}^{2})}{α_{+}^{2} (s + η - 2)} . \end{matrix}$ (4.10) Since $1 - η ⩽ 1 - s$ we have $ψ (s, η) ⩽ 3 (1 - s) .$ (4.11) Taylor expanding $W^{'}$ around $s = 1$ , we have $2 W^{'} (ξ) + 2 α_{+}^{2} (1 - ξ) - β_{+} {(1 - ξ)}^{2} = \frac{W^{''''} (ζ)}{6}$ for a suitable $ζ \in (ξ, 1)$ . Hence $R (s, η) = \frac{O ({(1 - ξ)}^{3})}{2 - (s + η)} = O ({(1 - s)}^{2}) .$ It then certainly follows from (4.9) $\frac{1}{\sqrt{2 W (s) - 2 W (η)}} = \frac{1 + O (1 - s)}{α_{+} \sqrt{{(1 - s)}^{2} - {(1 - η)}^{2}}} .$ Therefore, for $s < η$ and as $(s, η) \to (1^{-}, 1^{-})$ , $\frac{1}{\sqrt{2 W (s) - 2 W (η)}} - \frac{1}{α_{+} \sqrt{{(1 - s)}^{2} - {(1 - η)}^{2}}} = \frac{O (1 - s)}{\sqrt{{(1 - s)}^{2} - {(1 - η)}^{2}}} .$ (4.12) For $η \in (0, 1)$ sufficiently close to 1 we have $\begin{array}{rclr} D_{+} (η) & = & \int_{0}^{η} (\frac{1}{\sqrt{2 W (s) - 2 W (η)}} - \frac{1}{α_{+} \sqrt{{(1 - s)}^{2} - {(1 - η)}^{2}}}) d s \\ = & η \int_{0}^{1} (\frac{1}{\sqrt{2 W (s η) - 2 W (η)}} - \frac{1}{α_{+} \sqrt{{(1 - s η)}^{2} - {(1 - η)}^{2}}}) d s . & (4.13) \end{array}$ From (4.12) applied with $s η$ in place of s it follows $\begin{array}{rcl} | \frac{1}{\sqrt{2 W (s η) - 2 W (η)}} - \frac{1}{α_{+} \sqrt{{(1 - s η)}^{2} - {(1 - η)}^{2}}} | & ⩽ & \frac{C (1 - s η)}{\sqrt{{(1 - s η)}^{2} - {(1 - η)}^{2}}} \\ = & \frac{C}{\sqrt{1 - {((1 - η) / (1 - s η))}^{2}}} \end{array}$ for a suitable absolute positive constant C. Since the function $η \in (0, 1) \to \frac{1}{\sqrt{1 - {((1 - η) / (1 - s η))}^{2}}}$ is decreasing, the integrands on the right member of (4.13) are equiintegrable. Thus, by Lebesgue’s dominated convergence theorem, we can pass to the limit in (4.13) as $η \to 1^{-}$ and obtain $lim_{η \to 1^{-}} D_{+} (t) = \int_{0}^{1} (\frac{1}{\sqrt{2 W (s)}} - \frac{1}{α_{+} \sqrt{{(1 - s)}^{2}}}) d s = c_{+} . □$ (4.14)

Lemma 4.2 (Expansion of

Q_{+}

at first order).

Let $Q_{+}$ be the function defined in (4.1). Then $Q_{+} (η) = \frac{1}{α_{+}} log (\frac{1}{1 - η}) + \frac{log 2}{α_{+}} + c_{+} + o (1) as η \to 1^{-} .$ (4.15)

Proof.

Using (4.4) and (4.5) we have $Q_{+} (η) = \frac{1}{α_{+}} log (\frac{2}{1 - η}) + D_{+} (1) + o (1) as η \to 1^{-},$ (4.16) and the assertion follows from (4.8). □

Lemma 4.3 (Expansion of

L_{+}

at first order).

Let $L_{+}$ be the function defined in (4.6). Then $\begin{array}{rclr} L_{+} (η) & = & σ_{+} - \frac{α_{+}}{2} {(1 - η)}^{2} log (\frac{1}{1 - η}) \\ - \frac{α_{+}}{2} (log (2 κ_{+}) + \frac{1}{2}) {(1 - η)}^{2} + o ({(1 - η)}^{2}) as η \to 1^{-} . & (4.17) \end{array}$

Proof.
Using de l’Hôpital theorem, (4.7), (2.2) and (4.15), we compute $\begin{array}{rcl} lim_{η \to 1^{-}} \frac{L_{+} (η) - σ_{+} + (α_{+} / 2) {(1 - η)}^{2} log (1 / (1 - η))}{{(1 - η)}^{2}} \\ = \frac{1}{2} lim_{η \to 1^{-}} \frac{W^{'} (η) Q_{+} (η) + α_{+} (1 - η) log (1 / (1 - η)) - (α_{+} / 2) (1 - η)}{1 - η} \\ = \frac{1}{2} lim_{η \to 1^{-}} (- α_{+}^{2} (1 - η) [\frac{1}{α_{+}} log (\frac{1}{1 - η}) + \frac{log 2}{α_{+}} + c_{+}] \\ + α_{+} (1 - η) log (\frac{1}{1 - η}) - \frac{α_{+}}{2} (1 - η)) \\ / (1 - η) \\ = - \frac{α_{+}}{2} [log 2 + α_{+} c_{+} + \frac{1}{2}] . \end{array}$ Then formula (4.17) follows, recalling also (3.10). □

We shall use expansions (4.15) and (4.17) in formulas (6.13) and (6.16) below.
5. Two useful lemmas

In this section we prove two useful lemma, which are preliminary for the results of Section 6.

Lemma 5.1 (The functions $z_{ε}$ ).

Let $v \in BV (T; {\pm 1})$ be a function with $N (v) > 0$ . For any $k = 1, \dots, N (v)$ with k even, suppose that $v = - 1$ in $(x_{k - 1} (v), x_{k} (v))$ and $v = 1$ in $(x_{k} (v), x_{k + 1} (v))$ . For any $k = 1, \dots, N (v)$ , let $(x_{k}^{ε}) \subset T$ be a sequence of points converging to $x_{k} (v)$ as $ε \to 0^{+}$ , where we set $k_{N (v) + 1}^{ε} : = x_{1}^{ε}$ . Let $s_{0} \in (- 1, 0),$ (5.1) and define $A_{k}^{ε} (s_{0}) : = {z \in H^{1} (x_{k}^{ε}, x_{k + 1}^{ε}) : z (x_{k}^{ε}) = 0, z (x_{k + 1}^{ε}) = 0, {(- 1)}^{k} z (x) ⩾ s_{0} for any x \in (x_{k}^{ε}, x_{k + 1}^{ε})} .$ Then there exists a function $z_{ε} \in H^{1} (T; (- 1, 1))$ with the following properties:

for any $k = 1, \dots, N (v)$ $F_{ε} (z_{ε}, (x_{k}^{ε}, x_{k + 1}^{ε})) = min_{z \in A_{k}^{ε} (s_{0})} F_{ε} (z, (x_{k}^{ε}, x_{k + 1}^{ε}));$ (5.2)

there is a positive constant C depending only on W and v such that $sup_{ε \in (0, 1)} F_{ε} (z_{ε}) ⩽ C;$ (5.3)

for any $k = 1, \dots, N (v)$ we have $z_{ε} \in C^{1, 1} ([x_{k}^{ε}, x_{k + 1}^{ε}])$ ;

for any $k = 1, \dots, N (v)$ we have that, for $ε \in (0, 1)$ sufficiently small, $z_{ε} \in C^{\infty} (x_{k}^{ε}, x_{k + 1}^{ε})$ is a classical solution to $\{\begin{matrix} - ε z_{ε}^{″} + ε^{- 1} W^{'} (z_{ε}) = 0 in (x_{k}^{ε}, x_{k + 1}^{ε}), \\ z_{ε} (x_{k}^{ε}) = z_{ε} (x_{k + 1}^{ε}) = 0, \\ z_{ε} > 0 in (x_{k}^{ε}, x_{k + 1}^{ε}) if k is even, \\ z_{ε} < 0 in (x_{k}^{ε}, x_{k + 1}^{ε}) if k is odd . \end{matrix}$ (5.4) Moreover, $z_{k}^{ε}$ is even with respect to the mid point of $(x_{k}^{ε}, x_{k + 1}^{ε})$ ;

for any $k = 1, \dots, N (v)$ $lim_{ε \to 0^{+}} max_{x \in [x_{k}^{ε}, x_{k + 1}^{ε}]} | z_{ε} (x) | = 1 .$ (5.5)

Proof.

Given $k = 1, \dots, N (v)$ , the minimum problem on the right-hand side of (5.2) has a solution $z_{k}^{ε}$ by direct methods. Hence,4

⁴

If $a < b < c$ , $u \in H^{1} (a, b)$ , $v \in H^{1} (b, c)$ , and $u (b) = v (b)$ , then the function w defined as $w : = u$ in $(a, b)$ and $w : = v$ in $(b, c)$ , belongs to $H^{1} (a, c)$ .

setting

z_{ε} : = z_{k}^{ε} on [x_{k}^{ε}, x_{k + 1}^{ε}], k = 1, \dots, N (v),

we have that

z_{ε}

satisfies (i); note that, by truncating with the constants

- 1

and 1, we can suppose that

z_{ε} (x) \in [- 1, 1]

for any

x \in T

Assertion (ii) follows by comparing $F_{ε} (z_{ε})$ with the value of $F_{ε}$ , on each interval $(x_{k}^{ε}, x_{k + 1}^{ε})$ , of a suitable competitor. If $x_{k + 1}^{ε} - x_{k}^{ε} ⩽ 2 ε$ we choose the competitor to be identically zero; if $x_{k + 1}^{ε} - x_{k}^{ε} > 2 ε$ , for k even (resp. k odd) the competitor takes values in $[0, 1]$ (resp. in $[- 1, 0]$ ) and grows (resp. decreases) linearly from 0 to 1 (resp. from 0 to $- 1$ ) in $(x_{k}^{ε}, x_{k}^{ε} + ε)$ , it is 1 (resp. $- 1$ ) in $(x_{k}^{ε} + ε, x_{k + 1}^{ε} - ε)$ , and then decreases (resp. grows) to 0 (resp. to $- 1$ ) in $(x_{k + 1}^{ε} - ε, x_{k + 1}^{ε})$ .

Let us show (iii) and (iv). Without loss of generality, we fix k even. The minimality of $z_{ε}$ in $(x_{k}^{ε}, x_{k + 1}^{ε})$ entails $- ε z_{ε}^{″} + ε^{- 1} W^{'} (z_{ε}) ⩾ 0$ in the distributional sense in $(x_{k}^{ε}, x_{k + 1}^{ε})$ . It follows that $z_{ε}^{″} ⩽ ε^{- 2} {min}_{s \in [- 1, 1]} W^{'} (s)$ in the distributional sense in $(x_{k}^{ε}, x_{k + 1}^{ε})$ . Therefore $z_{ε}$ is semiconcave [7] in $(x_{k}^{ε}, x_{k + 1}^{ε})$ and, even more, it is semiconcave in $[x_{k}^{ε}, x_{k + 1}^{ε}]$ . As a consequence, the inequality $- ε z_{ε}^{″} + ε^{- 1} W^{'} (z_{ε}) ⩾ 0$ holds in $[x_{k}^{ε}, x_{k + 1}^{ε}]$ in the viscosity sense.

We also have that $z_{ε}$ is classical solution to $- ε z_{ε}^{″} + ε^{- 1} W^{'} (z_{ε}) = 0$ in the set ${z_{ε} > s_{0}} \cap (x_{k}^{ε}, x_{k + 1}^{ε})$ and, in particular, $- ε z_{ε}^{″} + ε^{- 1} W^{'} (z_{ε}) ⩽ 0$ in the viscosity sense in ${z_{ε} > s_{0}} \cap (x_{k}^{ε}, x_{k + 1}^{ε})$ , so that $z_{ε} \in C^{1, 1} ({z_{ε} > s_{0}} \cap (x_{k}^{ε}, x_{k + 1}^{ε})) \cap C^{\infty} ({z_{ε} > s_{0}} \cap (x_{k}^{ε}, x_{k + 1}^{ε})) .$ On ${z_{ε} = s_{0}} \cap (x_{k}^{ε}, x_{k + 1}^{ε})$ , the function $z_{ε}$ has a minimum, and therefore $- z_{ε}^{″} ⩽ 0$ in the viscosity sense. Coupled with the previous observation, we deduce $z_{ε} \in C^{1, 1} ([x_{k}^{ε}, x_{k + 1}^{ε}]) .$ The energy conservation implies that $ε \frac{{(z_{ε}^{'})}^{2}}{2} - ε^{- 1} W (z_{ε})$ is constant in any interval contained in ${z_{ε} > s_{0}} \cap (x_{k}^{ε}, x_{k + 1}^{ε})$ , therefore $ε \frac{{(z_{ε}^{'})}^{2}}{2} - ε^{- 1} W (z_{ε}) is a constant e (z_{k}^{ε}) in [x_{k}^{ε}, x_{k + 1}^{ε}] .$ (5.6) In particular $ε^{- 1} W (z_{ε}) ⩾ - e (z_{k}^{ε}) in [x_{k}^{ε}, x_{k + 1}^{ε}] .$ (5.7) We claim that $z_{ε} > s_{0} in [x_{k}^{ε}, x_{k + 1}^{ε}] .$ (5.8) Suppose by contradiction that ${z_{ε} = s_{0}} \cap (x_{k}^{ε}, x_{k + 1}^{ε}) \neq \emptyset$ . From (5.6) it follows that $- e (z_{k}^{ε}) = ε^{- 1} W (s_{0})$ because on the set ${z_{ε} = s_{0}}$ there holds $z_{ε}^{'} = 0$ . We deduce from (5.7) $F_{ε} (z_{ε}, (x_{k}^{ε}, x_{k + 1}^{ε})) ⩾ ε^{- 1} W (s_{0}) (x_{k + 1}^{ε} - x_{k}^{ε}) > C,$ for $ε \in (0, 1)$ sufficiently small depending only on v and $s_{0}$ , in contradiction with (5.3). We conclude that ${z_{ε} = s_{0}} \cap (x_{k}^{ε}, x_{k + 1}^{ε}) = \emptyset$ , and this proves our claim (5.8). Notice that the same argument shows that $z_{ε}$ cannot have critical points in ${z_{ε} < 0} \cap (x_{k}^{ε}, x_{k + 1}^{ε})$ , hence in particular $z_{ε} > 0 in (x_{k}^{ε}, x_{k + 1}^{ε}) .$

The proof of the validity of the ordinary differential equation in (5.4) then follows, and hence by uniqueness $z_{ε} (x) \in (- 1, 1)$ for any $x \in T$ .

Let us show that $z_{k}^{ε}$ is even with respect to the mid point of $(x_{k}^{ε}, x_{k + 1}^{ε})$ . Let $\overline{x} \in (x_{k}^{ε}, x_{k + 1}^{ε})$ be a point where $z_{k}^{ε}$ takes the maximum value in $[x_{k}^{ε}, x_{k + 1}^{ε}]$ . Observe that ${\tilde{z}}_{k}^{ε} (x) : = z_{k}^{ε} (2 \overline{x} - x)$ solves the ordinary differential equation in (5.4), with ${\tilde{z}}_{k}^{ε} (\overline{x}) = z_{k}^{ε} (\overline{x})$ and ${\tilde{z}}_{k}^{ε'} (\overline{x}) = z_{k}^{ε'} (\overline{x}) = 0$ . Hence by uniqueness ${\tilde{z}}_{k}^{ε} = z_{k}^{ε}$ . If by contradiction $\overline{x}$ is not the mid point of $(x_{k}^{ε}, x_{k + 1}^{ε})$ , we have that $z_{k}^{ε}$ vanishes somewhere in $(x_{k}^{ε}, x_{k + 1}^{ε})$ , which is impossible, because $z_{k}^{ε} > 0$ by (5.4).

Assertion (v) follows, because contradicting (5.5) would contradict estimate (5.3). □

Note that assertions (ii)–(v) are valid independently of $s_{0}$ ; we shall make use of $s_{0}$ in the second case of the proof of the next lemma. We need the following preliminary observation. Let $v \in BV (T; {\pm 1})$ and $(v_{ε}) \subset H^{1} (T)$ be a sequence converging to v in $L^{1} (T)$ as $ε \to 0^{+}$ . The continuity of $v_{ε}$ and the convergence of $(v_{ε})$ to v imply that, for any $k = 1, \dots, N (v)$ , there exists a sequence $(x_{k}^{ε}) \subset T$ of points converging to $x_{k} (v)$ , such that $v_{ε} (x_{k}^{ε}) = 0,$ (5.9) where $x_{N (v) + 1}^{ε} : = x_{1}^{ε}$ . Lemma 5.2 (Action comparison between $v_{ε}$ and $z_{ε}$ ).

Let v be as in Lemma 5.1. Let $(v_{ε}) \subset H^{1} (T)$ be a sequence converging to v in $L^{1} (T)$ as $ε \to 0^{+}$ . For any $k = 1, \dots, N (v)$ , select a sequence $(x_{k}^{ε}) \subset T$ of points converging to $x_{k} (v)$ such that $v_{ε} (x_{k}^{ε}) = 0$ , where we have set $x_{N (v) + 1}^{ε} : = x_{1}^{ε}$ . With $s_{0}$ as in (5.1), let $(z_{ε})$ be the sequence of functions given by Lemma 5.1. Then $F_{ε} (v_{ε}, (x_{k}^{ε}, x_{k + 1}^{ε})) ⩾ F_{ε} (z_{ε}, (x_{k}^{ε}, x_{k + 1}^{ε})), k = 1, \dots, N (v)$ (5.10) for $ε \in (0, 1)$ small enough.

Proof.

Without loss of generality, let us fix k even. We divide the proof into two cases. Case 1.

$v_{ε} ⩾ s_{0}$ in $(x_{k}^{ε}, x_{k + 1}^{ε})$ .

In this case we have that $v_{ε} \in A_{k}^{ε} (s_{0})$ , and (5.10) follows by the minimality of $z_{ε}$ (see (5.2)). Case 2.

Suppose that $v_{ε} (\overline{x}) < s_{0}$ for some $\overline{x} \in (x_{k}^{ε}, x_{k + 1}^{ε})$ .

We have $\begin{array}{rcl} F_{ε} (v_{ε}, (x_{k}^{ε}, x_{k + 1}^{ε})) & ⩾ & \int_{(x_{k}^{ε}, x_{k + 1}^{ε})} \sqrt{2 W (v_{ε})} | v_{ε}^{'} | d x = \int_{(x_{k}^{ε}, x_{k + 1}^{ε})} | ϕ {(v_{ε})}^{'} | d x \\ ⩾ & (ϕ (0) - ϕ (m_{k}^{ε})) + (ϕ (M_{k}^{ε}) - ϕ (m_{k}^{ε})) + (ϕ (M_{k}^{ε}) - ϕ (0)) \\ = & 2 (ϕ (M_{k}^{ε}) - ϕ (m_{k}^{ε})), \end{array}$ where ϕ is defined in (2.4), and $M_{k}^{ε} : = max {v_{ε} (x) : x \in [x_{k}^{ε}, x_{k + 1}^{ε}]} > m_{k}^{ε} : = min {v_{ε} (x) : x \in [x_{k}^{ε}, x_{k + 1}^{ε}]} .$ Since ϕ is strictly increasing and $m_{k}^{ε} < s_{0}$ , we deduce $F_{ε} (v_{ε}, (x_{k}^{ε}, x_{k + 1}^{ε})) ⩾ 2 (ϕ (M_{k}^{ε}) - ϕ (s_{0})) ⩾ 2 (ϕ (1) - ϕ (s_{0})) + o (1)$ (5.11) as $ε \to 0^{+}$ , where in the last inequality we have used that ${lim}_{ε \to 0^{+}} {∥ v_{ε} - 1 ∥}_{L^{1} (x_{k}^{ε}, x_{k + 1}^{ε})} = 0$ .

For any $k = 1, \dots, N (v) - 1$ let now $d_{k}^{ε} : = x_{k + 1}^{ε} - x_{k}^{ε}$ and $d_{N (v)}^{ε} : = 1 - (x_{N (v)}^{ε} - x_{1}^{ε})$ , so that ${lim}_{ε \to 0^{+}} d_{k}^{ε} = d_{k} (v)$ .

Define ${\tilde{z}}_{k}^{ε} (x) : = \{\begin{matrix} γ (\frac{x - x_{k}^{ε}}{ε}) & if x \in (x_{k}^{ε}, x_{k}^{ε} + \frac{d_{k}^{ε}}{2}), \\ γ (\frac{x_{k + 1}^{ε} - x}{ε}) & if x \in (x_{k}^{ε} + \frac{d_{k}^{ε}}{2}, x_{k + 1}^{ε}), \end{matrix}$ where γ solves (2.9). We have ${\tilde{z}}_{k}^{ε} \in A_{k}^{ε} (s_{0})$ and, as $ε \to 0^{+}$ , $F_{ε} ({\tilde{z}}_{k}^{ε}, (x_{k}^{ε}, x_{k + 1}^{ε})) = 2 (ϕ (max_{x \in [x_{k}^{ε}, x_{k + 1}^{ε}]} {\tilde{z}}_{k}^{ε} (x)) - ϕ (0)) = 2 (ϕ (1) - ϕ (0)) + o (1) .$ (5.12) In addition, by minimality, $F_{ε} ({\tilde{z}}_{k}^{ε}, (x_{k}^{ε}, x_{k + 1}^{ε})) ⩾ F_{ε} (z_{ε}, (x_{k}^{ε}, x_{k + 1}^{ε})) .$ (5.13) Then (5.10) follows from (5.11), (5.12) and (5.13) provided $ε > 0$ is sufficiently small since, being ϕ strictly increasing and $s_{0} \in (- 1, 0)$ , $\begin{array}{rcl} F_{ε} (v_{ε}, (x_{k}^{ε}, x_{k}^{ε})) & ⩾ & 2 (ϕ (1) - ϕ (s_{0})) + o (1) > 2 (ϕ (1) - ϕ (0)) + o (1) \\ = & F_{ε} ({\tilde{z}}_{k}^{ε}, (x_{k}^{ε}, x_{k + 1}^{ε})) ⩾ F_{ε} (z_{k}^{ε}, (x_{k}^{ε}, x_{k + 1}^{ε})) . □ \end{array}$

6. First order estimate for

F_{ε}

In this section we prove the first order expansion for $F_{ε}$ , in the sense specified by Theorems 6.1 and 6.5.

Theorem 6.1 (First order estimate from below).

Suppose that assumptions ( W1)–(W3) hold. Let $(v_{ε}) \subset H^{1} (T)$ be a sequence converging in $L^{1} (T)$ to a nonconstant function $v \in BV (T; {\pm 1})$ . Then, for any $k = 1, \dots, N (v)$ , there exists a sequence $(d_{k}^{ε})$ satisfying ${lim}_{ε \to 0^{+}} d_{k}^{ε} = d_{k} (v)$ such that $\begin{array}{rclr} F_{ε} (v_{ε}) & ⩾ & N (v) σ - α_{+} κ_{+}^{2} \sum_{k = 1}^{N (v)} e^{- α_{+} (d_{k}^{ε} / ε)} - α_{-} κ_{-}^{2} \sum_{k = 1}^{N (v)} e^{- α_{-} (d_{k}^{ε} / ε)} \\ + o (\sum_{k = 1}^{N (v)} e^{- α_{+} (d_{k}^{ε} / ε)}) + o (\sum_{k = 1}^{N (v)} e^{- α_{-} (d_{k}^{ε} / ε)}) as ε \to 0^{+} . & (6.1) \end{array}$

Proof.
Without loss of generality, we can assume $N (v) ⩾ 2$ , and that $v = - 1$ in $(x_{k - 1} (v), x_{k} (v))$ and $v = 1$ in $(x_{k} (v), x_{k + 1} (v))$ for any $k = 1, \dots, N (v)$ , k even. For any $k = 1, \dots, N (v)$ select a sequence $(x_{k}^{ε})$ of points of $T$ satisfying ${lim}_{ε \to 0^{+}} x_{k}^{ε} = x_{k} (v)$ and equality (5.9), where $x_{N (v) + 1}^{ε} : = x_{1}^{ε}$ .

Now, let $d_{k}^{ε}$ be defined as $\{\begin{matrix} d_{k}^{ε} : = x_{k + 1}^{ε} - x_{k}^{ε}, k = 1, \dots, N (v) - 1, \\ d_{N (v)}^{ε} : = 1 - (x_{N (v)}^{ε} - x_{1}^{ε}) = : d_{0}^{ε}, \end{matrix}$ (6.2) and set $I_{k}^{ε} (x_{k}^{ε}) : = (x_{k}^{ε} - \frac{d_{k - 1}^{ε}}{2}, x_{k}^{ε} + \frac{d_{k}^{ε}}{2}) .$ From inequality (5.10) of Lemma 5.2 it follows $\begin{array}{rclr} F_{ε} (v_{ε}) & = & \sum_{k = 1}^{N (v)} F_{ε} (v_{ε}, (x_{k}^{ε}, x_{k + 1}^{ε})) ⩾ \sum_{k = 1}^{N (v)} F_{ε} (z_{ε}, (x_{k}^{ε}, x_{k + 1}^{ε})) \\ = & F_{ε} (z_{ε}) = \sum_{k = 1}^{N (v)} F_{ε} (z_{ε}, I_{k}^{ε} (x_{k}^{ε})) & (6.3) \end{array}$ for $ε \in (0, 1)$ small enough. With the change of variable $x = ε y + x_{k}^{ε}$ we get $F_{ε} (z_{ε}, I_{k}^{ε} (x_{k}^{ε})) = \int_{\frac{1}{ε} I_{k}^{ε} (0)} (\frac{ε^{2}}{2} {(z_{ε}^{'} (ε y + x_{k}^{ε}))}^{2} + W (z_{ε} (ε y + x_{k}^{ε}))) d y,$ (6.4) where $\frac{1}{ε} I_{k}^{ε} (0) = (\frac{- d_{k - 1}^{ε}}{2 ε}, \frac{d_{k}^{ε}}{2 ε}) .$ (6.5) Let $w_{k}^{ε} \in H^{1} (\frac{1}{ε} I_{k}^{ε} (0))$ be the function defined as $w_{k}^{ε} (y) : = z_{ε} (ε y + x_{k}^{ε}), y \in \frac{1}{ε} I_{k}^{ε} (0),$ (6.6) where we set $w_{0}^{ε} : = w_{N (v)}^{ε}$ . We deduce $F_{ε} (z_{ε}, I_{k}^{ε} (x_{k}^{ε})) = \int_{\frac{1}{ε} I_{k}^{ε} (0)} (\frac{1}{2} {({w_{k}^{ε}}^{'} (y))}^{2} + W (w_{k}^{ε} (y))) d y = F (w_{k}^{ε}, \frac{1}{ε} I_{k}^{ε} (0)),$ (6.7) where $F$ is defined in (2.8). Hence, from (6.3), $\begin{array}{rclr} F_{ε} (v_{ε}) & ⩾ & \sum_{k = 1}^{N (v)} F (w_{k}^{ε}, \frac{1}{ε} I_{k}^{ε} (0)) \\ = & \sum_{k \in I_{+} (v)} F (w_{k}^{ε}, \frac{1}{ε} I_{k}^{ε} (0)) + \sum_{k \in I_{-} (v)} F (w_{k}^{ε}, \frac{1}{ε} I_{k}^{ε} (0)) & (6.8) \end{array}$ for $ε \in (0, 1)$ small enough. Observe from (5.4) that $w_{k}^{ε}$ solves $\{\begin{matrix} - {w_{k}^{ε}}^{″} + W^{'} (w_{k}^{ε}) = 0 in (\frac{- d_{k - 1}^{ε}}{2 ε}, \frac{d_{k}^{ε}}{2 ε}) ∖ {0}, \\ w_{k}^{ε} (0) = 0 . \end{matrix}$ (6.9) Moreover, from (5.5) we get $lim_{ε \to 0^{+}} | w_{k}^{ε} (\frac{- d_{k - 1}^{ε}}{2 ε}) | = lim_{ε \to 0^{+}} | w_{k}^{ε} (\frac{d_{k}^{ε}}{2 ε}) | = 1 .$ (6.10) Define $e (w_{k - 1}^{ε})$ and $e (w_{k}^{ε})$ as the (conserved) energy densities of $w_{k}^{ε}$ in $(- \frac{d_{k - 1}^{ε}}{2 ε}, 0)$ and $(0, \frac{d_{k}^{ε}}{2 ε})$ respectively, namely $\begin{matrix} e (w_{k - 1}^{ε}) : = \frac{{({w_{k}^{ε}}^{' -} (0))}^{2}}{2} - W (0) = - W (w_{k}^{ε} (- \frac{d_{k - 1}^{ε}}{2 ε})) < 0, \\ e (w_{k}^{ε}) : = \frac{{({w_{k}^{ε}}^{' +} (0))}^{2}}{2} - W (0) = - W (w_{k}^{ε} (\frac{d_{k}^{ε}}{2 ε})) < 0, \end{matrix}$ (6.11) where ${w_{k}^{ε}}^{' -} (0)$ (resp. ${w_{k}^{ε}}^{' +} (0)$ ) stands for the left (resp. right) derivative of $w_{k}^{ε}$ at 0. Set also $e (w_{0}^{ε}) : = e (w_{N (v)}^{ε})$ . We have, for $k \in I_{+} (v)$ , ${w_{k}^{ε}}^{'} = \{\begin{matrix} \sqrt{2 W (w_{k}^{ε}) + 2 e (w_{k - 1}^{ε})} & in (\frac{- d_{k - 1}^{ε}}{2 ε}, 0), \\ \sqrt{2 W (w_{k}^{ε}) + 2 e (w_{k}^{ε})} & in (0, \frac{d_{k}^{ε}}{2 ε}), \\ w_{k}^{ε} (0) = 0 . \end{matrix}$ (6.12)

Hence $\frac{d_{k}^{ε}}{2 ε} = \int_{0}^{w_{k}^{ε} (d_{k}^{ε} / 2 ε)} \frac{1}{\sqrt{2 W (s) + 2 e (w_{k}^{ε})}} d s .$

From (6.11) and the expression of $Q_{+}$ in (4.1), we get $\frac{d_{k}^{ε}}{2 ε} = Q_{+} (w_{k}^{ε} (\frac{d_{k}^{ε}}{2 ε})) .$ (6.13)

From (6.13), (3.10) and (4.15) we deduce, as $ε \to 0^{+}$ , $log (\frac{1}{1 - w_{k}^{ε} (d_{k}^{ε} / 2 ε)}) = α_{+} \frac{d_{k}^{ε}}{2 ε} - log (2 κ_{+}) + o (1),$ (6.14) and therefore5
⁵
From (6.15) it follows ${lim}_{ε \to 0^{+}} \frac{{(1 - w_{k}^{ε} (d_{k}^{ε} / 2 ε))}^{2}}{e^{- α_{+} (d_{k}^{ε} / ε)}} = 4 κ_{+}^{2}$ , a formula also proven in [8, Proposition 3.4].

$1 - w_{k}^{ε} (\frac{d_{k}^{ε}}{2 ε}) = 2 κ_{+} e^{- α_{+} (d_{k}^{ε} / 2 ε) + o (1)} as ε \to 0^{+} .$ (6.15) Adding and subtracting the term $W (w_{k}^{ε} (\frac{d_{k}^{ε}}{2 ε}))$ inside the integral, and taking advantage of (6.12), for $k \in I_{+} (v)$ we write $\begin{array}{rclr} F (w_{k}^{ε}, (0, \frac{d_{k}^{ε}}{2 ε})) \\ = \int_{0}^{d_{k}^{ε} / 2 ε} (\frac{{({w_{k}^{ε}}^{'})}^{2}}{2} + W (w_{k}^{ε}) - W (w_{k}^{ε} (\frac{d_{k}^{ε}}{2 ε}))) d y + \frac{d_{k}^{ε}}{2 ε} W (w_{k}^{ε} (\frac{d_{k}^{ε}}{2 ε})) \\ = \int_{0}^{d_{k}^{ε} / 2 ε} \sqrt{2 W (w_{k}^{ε} (y)) - 2 W (w_{k}^{ε} (\frac{d_{k}^{ε}}{2 ε}))} {w_{k}^{ε}}^{'} (y) d y + \frac{d_{k}^{ε}}{2 ε} W (w_{k}^{ε} (\frac{d_{k}^{ε}}{2 ε})) \\ = \int_{0}^{w_{k}^{ε} (d_{k}^{ε} / 2 ε)} \sqrt{2 W (s) - 2 W (w_{k}^{ε} (\frac{d_{k}^{ε}}{2 ε}))} d s + \frac{d_{k}^{ε}}{2 ε} W (w_{k}^{ε} (\frac{d_{k}^{ε}}{2 ε})) \\ = L_{+} (w_{k}^{ε} (\frac{d_{k}^{ε}}{2 ε})) + \frac{d_{k}^{ε}}{2 ε} W (w_{k}^{ε} (\frac{d_{k}^{ε}}{2 ε})), & (6.16) \end{array}$ where we recall that $L_{+}$ is defined in (4.6).

Substituting (4.17) and (2.2) into (6.16) we deduce, using also (6.14) and (6.15), $\begin{array}{rcl} F (w_{k}^{ε}, (0, \frac{d_{k}^{ε}}{2 ε})) \\ = σ_{+} - \frac{α_{+}}{2} {(1 - w_{k}^{ε} (\frac{d_{k}^{ε}}{2 ε}))}^{2} log (\frac{1}{1 - w_{k}^{ε} (d_{k}^{ε} / 2 ε)}) \\ - \frac{α_{+}}{2} (log (2 κ_{+}) + \frac{1}{2}) {(1 - w_{k}^{ε} (\frac{d_{k}^{ε}}{2 ε}))}^{2} \\ + \frac{d_{k}^{ε}}{2 ε} \frac{α_{+}^{2}}{2} {(1 - w_{k}^{ε} (\frac{d_{k}^{ε}}{2 ε}))}^{2} + o ({(1 - w_{k}^{ε} (\frac{d_{k}^{ε}}{2 ε}))}^{2}) \\ = σ_{+} - \frac{α_{+}}{2} (log (2 κ_{+}) + \frac{1}{2} - log (2 κ_{+})) {(1 - w_{k}^{ε} (\frac{d_{k}^{ε}}{2 ε}))}^{2} \\ + o ({(1 - w_{k}^{ε} (\frac{d_{k}^{ε}}{2 ε}))}^{2}) \\ = σ_{+} - α_{+} κ_{+}^{2} e^{- α_{+} (d_{k}^{ε} / ε)} + o (e^{- α_{+} (d_{k}^{ε} / ε)}) \end{array}$ as $ε \to 0^{+}$ .

With similar arguments one can prove that $F (w_{k}^{ε}, (- \frac{d_{k - 1}^{ε}}{2 ε}, 0)) = σ_{-} - α_{-} κ_{-}^{2} e^{- α_{-} (d_{k - 1}^{ε} / ε)} + o (e^{- α_{-} (d_{k - 1}^{ε} / ε)}) .$ Hence, for $k \in I_{+} (v)$ we get $\begin{array}{rclr} F (w_{k}^{ε}, \frac{1}{ε} I_{k}^{ε} (0)) = σ - α_{+} κ_{+}^{2} e^{- α_{+} (d_{k}^{ε} / ε)} - α_{-} κ_{-}^{2} e^{- α_{-} (d_{k - 1}^{ε} / ε)} + o (e^{- α_{+} (d_{k}^{ε} / ε)}) + o (e^{- α_{-} (d_{k - 1}^{ε} / ε)}) . & (6.17) \end{array}$ Similarly, for $k \in I_{-} (v)$ , we have $\begin{array}{rclr} F (w_{k}^{ε}, \frac{1}{ε} I_{k}^{ε} (0)) = σ - α_{-} κ_{-}^{2} e^{- α_{-} (d_{k}^{ε} / ε)} - α_{+} κ_{+}^{2} e^{- α_{+} (d_{k - 1}^{ε} / ε)} + o (e^{- α_{-} (d_{k}^{ε} / ε)}) + o (e^{- α_{+} (d_{k - 1}^{ε} / ε)}) . & (6.18) \end{array}$ From (6.8), (6.17) and (6.18) the assertion of the theorem follows. □
Remark 6.2 (W even).

When W is even, in order to prove Theorem 6.1 there is no need to introduce $s_{0}$ as in (5.1), and there is no need to use Lemmas 5.1 and 5.2. Indeed, if W is even, we can define $z_{k}^{ε}$ a solution to (5.2) where $s_{0}$ is replaced by 0, and we can directly prove inequality (5.10), since $F_{ε} (v_{ε}, (x_{k}^{ε}, x_{k + 1}^{ε})) = F_{ε} (| v_{ε} |, (x_{k}^{ε}, x_{k + 1}^{ε})) ⩾ F_{ε} (z_{k}^{ε}, (x_{k}^{ε}, x_{k + 1}^{ε})) .$

Remark 6.3.
As soon as $d_{k} (v) \neq d_{h} (v)$ , the corresponding infinitesimals $e^{- α_{\pm} (d_{k}^{ε} / ε)}$ , $e^{- α_{\pm} (d_{h}^{ε} / ε)}$ on the right-hand side of (6.1) are not comparable. It may happen that the error on a term of the sum, say $e^{- α_{\pm} (d_{k}^{ε} / ε)}$ , is larger then another term of the sum, say $e^{- α_{\pm} (d_{h}^{ε} / ε)}$ . A rougher estimate than (6.1) is obtained by replacing the terms $o (\sum_{k = 1}^{N (v)} e^{- α_{\pm} (d_{k}^{ε} / ε)})$ with $o (e^{- α_{\pm} {min}_{k = 1, \dots, N (v)} d_{k}^{ε} / ε}) .$
Corollary 6.4.
Let $(v_{ε}) \subset H^{1} (T)$ be a sequence converging in $L^{1} (T)$ to a nonconstant function $v \in BV (T; {\pm 1})$ . Then, for any $d \in (0, min {d_{k} (v) : k = 1, \dots, N (v)})$ and any $C_{+} > 0$ , $C_{-} > 0$ we have $F_{ε} (v_{ε}) ⩾ N (v) σ - C_{+} e^{- α_{+} (d / ε)} - C_{-} e^{- α_{-} (d / ε)} + o (e^{- α_{+} (d / ε)}) + o (e^{- α_{-} (d / ε)})$ (6.19) as $ε \to 0^{+}$ .
Theorem 6.5 (First order estimate from above).

Suppose that assumptions ( W1)–(W3) hold. Let $v \in BV (T; {\pm 1})$ be a nonconstant function. Then there exists a sequence $(v_{ε}) \subset H^{1} (T)$ converging to v in $L^{1} (T)$ and satisfying the inequality $\begin{array}{rclr} F_{ε} (v_{ε}) & ⩽ & N (v) σ - α_{+} κ_{+}^{2} \sum_{k = 1}^{N (v)} e^{- α_{+} (d_{k} (v) / ε)} - α_{-} κ_{-}^{2} \sum_{k = 1}^{N (v)} e^{- α_{-} (d_{k} (v) / ε)} \\ + o (\sum_{k = 1}^{N (v)} e^{- α_{+} (d_{k} (v) / ε)}) + o (\sum_{k = 1}^{N (v)} e^{- α_{-} (d_{k} (v) / ε)}) as ε \to 0^{+} . & (6.20) \end{array}$

Proof.
By standard arguments, it is sufficient to prove the statement for a function v having only two jumps $x_{1} (v) < x_{2} (v)$ . Let $d_{1} (v) : = x_{2} (v) - x_{1} (v)$ and $d_{2} (v) : = 1 - d_{1} (v) = : d_{0} (v)$ . Without loss of generality, we can assume $v = \{\begin{matrix} - 1 & in (x_{1} (v), x_{2} (v)), \\ 1 & in (x_{2} (v), x_{1} (v)) . \end{matrix}$ Set $I_{1} (x_{1} (v)) : = (x_{1} (v) - \frac{d_{2} (v)}{2}, x_{1} (v) + \frac{d_{1} (v)}{2}), I_{2} (x_{2} (v)) : = (x_{2} (v) - \frac{d_{1} (v)}{2}, x_{2} (v) + \frac{d_{2} (v)}{2}) .$ Let $z_{1}^{ε}$ , $z_{2}^{ε}$ and $z_{ε}$ be as in Lemma 5.1 (with $N (v) = 2$ ) with the choice $x_{1}^{ε} : = x_{1} (v), x_{2}^{ε} : = x_{2} (v), ε \in (0, 1) .$ Write $\frac{1}{ε} I_{1} (0) : = (\frac{- d_{0} (v)}{2 ε}, \frac{d_{1} (v)}{2 ε}), (\frac{- d_{1} (v)}{2 ε}, \frac{d_{2} (v)}{2 ε}),$ and let $w_{1}^{ε} \in H^{1} (\frac{1}{ε} I_{1} (0))$ and $w_{2}^{ε} = w_{0}^{ε} \in H^{1} (\frac{1}{ε} I_{2} (0))$ be defined as in (6.6). Then $w_{1}^{ε}$ and $w_{2}^{ε}$ satisfy (6.11) and (6.12). We define $v_{ε} (x) : = \{\begin{matrix} w_{2}^{ε} (\frac{x_{2} (v) - x}{ε}) & if x \in I_{2} (x_{2} (v)), \\ w_{1}^{ε} (\frac{x - x_{1} (v)}{ε}) & if x \in I_{1} (x_{1} (v)) . \end{matrix}$ Then $v_{ε} \in H^{1} (T)$ , $(v_{ε})$ converges to v in $L^{1} (T)$ as $ε \to 0^{+}$ , and $F_{ε} (v_{ε}) = F_{ε} (v_{ε}, I_{1} (x_{1} (v))) + F_{ε} (v_{ε}, I_{2} (x_{2} (v))) .$ (6.21) With the change of variable $y = \frac{x - x_{2} (v)}{ε}$ , we have $F_{ε} (v_{ε}, I_{2} (x_{2} (v))) = F (w_{2}^{ε}, \frac{1}{ε} I_{2} (0))$ , and, as in (6.16), $F (w_{2}^{ε}, (0, \frac{d_{2} (v)}{2 ε})) = L_{+} (w_{2}^{ε} (\frac{d_{2} (v)}{2 ε})) + \frac{d_{2} (v)}{2 ε} W (w_{2}^{ε} (\frac{d_{2} (v)}{2 ε})) .$ (6.22) Then the proof follows along the same lines as the proof of Theorem 6.1. □
Remark 6.6.
With slight modifications in the proof of Theorem 6.5, one can show that for any sequence $(d_{k}^{ε})$ converging to $d_{k} (v)$ , there exists a sequence $(v_{ε}) \subset H^{1} (T)$ converging to v in $L^{1} (T)$ and satisfying the equality in (6.1) with $d_{k}^{ε} = d_{k} (v)$ for any $k = 1, \dots, N (v)$ .
Remark 6.7.
From Theorems 6.1 and 6.5 it follows that, given $γ > 0$ , for any $v \in BV (T; {\pm 1})$ with $N (v) > 0$ and for any sequence $(v_{ε}) \in H^{1} (T)$ such that $v_{ε} \to v$ in $L^{1} (T)$ as $ε \to 0^{+}$ , there holds $\underset{ε \to 0^{+}}{lim inf} \frac{F_{ε} (v_{ε}) - N (v) σ}{ε^{γ}} ⩾ 0,$ with the equality along a particular sequence. Hence the Γ-expansion of the functionals $F_{ε}$ in the sense of [5], whose zeroth-order is given by $N (\cdot) σ$ , contains no terms of order $ε^{γ}$ for any $γ > 0$ .

7. Γ-convergence

Throughout this short section, $N \in N$ and $m > 0$ are fixed, and we assume for simplicity that W is even. We set $α : = α_{-} = α_{+}$ (see (2.1)) and $κ : = κ_{-} = κ_{+}$ (see (3.10)). For any $ε \in (0, 1]$ we define the functionals $T_{ε}^{N, m} : L^{1} (T) \to (- \infty, + \infty]$ as $T_{ε}^{N, m} (v) : = e^{α (m / ε)} (F_{ε} (v) - N σ) .$ Observe that $T_{ε}^{N, m}$ may take negative values.

Remark 7.1.
Let $(v_{ε}) \subset H^{1} (T)$ be such that $sup_{ε \in (0, 1]} T_{ε}^{N, m} (v_{ε}) < + \infty .$ (7.1) Then ${sup}_{ε \in (0, 1]} F_{ε} (v_{ε}) < + \infty$ . Hence $(v_{ε})$ admits a (not relabeled) subsequence converging in $L^{1} (T)$ to a function $v \in BV (T; {\pm 1})$ , and $N (v) σ ⩽ Γ - L^{1} (T) \underset{ε \to 0^{+}}{lim inf} F_{ε} (v) ⩽ \underset{ε \to 0^{+}}{lim sup} F_{ε} (v_{ε}) ⩽ N σ,$ where the last inequality follows from (7.1). Hence $N ⩾ N (v) .$
Theorem 7.2 (First order Γ-limit).

Suppose that assumptions ( W1)–(W3) hold, and that in addition W is even. Then the sequence $(T_{ε}^{N, m})$ Γ- $L^{1} (T)$ -converges, as $ε \to 0^{+}$ , to the functional $T^{N, m} : L^{1} (T) \to [- \infty, + \infty]$ given by $T^{N, m} (v) = \{\begin{matrix} 0 & if v \in BV (T; {\pm 1}), N = N (v) and m < m (v), \\ - \infty & if v \in BV (T; {\pm 1}), N = N (v) and m ⩾ m (v), \\ + \infty & if v \in BV (T; {\pm 1}) and N < N (v), \\ - \infty & if v \in BV (T; {\pm 1}) and N > N (v), \\ + \infty & if v \in L^{1} (T) ∖ BV (T; {\pm 1}), \end{matrix}$ where for any $v \in BV (T; {\pm 1})$ , we have set $m (v) : = min {d_{k} (v) : k = 1, \dots, N (v)}$ .

Proof.
Set $T^{+} : = Γ - L^{1} (T) {lim sup}_{ε \to 0^{+}} T_{ε}^{N, m}$ and $T^{-} : = Γ - L^{1} (T) {lim inf}_{ε \to 0^{+}} T_{ε}^{N, m}$ , see [9]. Let $v \in L^{1} (T)$ , and let $(v_{ε}) \subset H^{1} (T)$ be a sequence satisfying (7.1) and converging to v in $L^{1} (T)$ . Then $v \in BV (T; {\pm 1})$ and $N ⩾ N (v)$ , so that $T^{-} (v) = + \infty$ if $either v \in L^{1} (T) ∖ BV (T; {\pm 1}) or v \in BV (T; {\pm 1}) and N < N (v),$ (7.2) and therefore $Γ - L^{1} (T) lim_{ε \to 0^{+}} T_{ε}^{N, m} (v) = + \infty if v satisfies (7.2) .$ We can assume from now on that $v \in BV (T; {\pm 1})$ . The continuity of $v_{ε}$ and the convergence of $(v_{ε})$ to v imply that there exist an infinitesimal sequence $(δ_{ε}) \subset (0, 1)$ and, for any $k = 1, \dots, N (v)$ , a sequence of points $(x_{k}^{ε}) \subset T$ , such that for any $ε \in (0, 1)$ , $| x_{k} (v) - x_{k}^{ε} | ⩽ δ_{ε},$ and (5.9) holds. From Theorem 6.1, (6.1), and $d_{k}^{ε} = x_{k + 1}^{ε} - x_{k}^{ε} ⩾ x_{k + 1} (v) - x_{k} (v) - 2 δ_{ε} ⩾ m (v) - 2 δ_{ε}$ , we have $F_{ε} (v_{ε}) ⩾ N (v) σ - 2 α κ^{2} # {k = 1, \dots, N (v) : d_{k} (v) = m (v)} e^{- α ((m (v) - 2 δ_{ε}) / ε)} + o (e^{- α ((m (v) - 2 δ_{ε}) / ε)})$ since the contribution due to the remaining jump points is of higher order. Moreover, $\begin{array}{rcl} T_{ε}^{N, m} (v_{ε}) & = & e^{α (m / ε)} (F_{ε} (v_{ε}) - N σ) \\ ⩾ & e^{α (m / ε)} (N (v) - N) σ - 2 α κ^{2} # {k = 1, \dots, N (v) : d_{k} (v) = m (v)} e^{α ((m - m (v) + 2 δ_{ε}) / ε)} \\ + o (e^{α ((m - m (v) + 2 δ_{ε}) / ε)}) . \end{array}$

Hence $N = N (v), m < m (v) \Rightarrow T^{-} (v) ⩾ 0 .$ (7.3)

If now $(v_{ε})$ denotes the sequence constructed in Theorem 6.5, we have $T_{ε}^{N, m} (v_{ε}) ⩽ e^{α (m / ε)} (N (v) - N) σ - 2 α κ^{2} \sum_{k = 1}^{N (v)} e^{α ((m - d_{k} (v)) / ε)} + o (\sum_{k = 1}^{N (v)} e^{α ((m - d_{k} (v)) / ε)}) .$ (7.4)

Therefore, for a v satisfying (7.3), we have ${lim}_{ε \to 0^{+}} T_{ε}^{N, m} (v_{ε}) = 0$ , hence $T^{+} (v) ⩽ 0$ , which coupled with (7.3) gives $N < N (v), m < m (v) \Rightarrow Γ - L^{1} (T) lim_{ε \to 0^{+}} T_{ε}^{N, m} (v) = 0 .$

If either $N > N (v)$ or $N = N (v)$ and $m > m (v)$ , from (7.4) it follows ${lim sup}_{ε \to 0^{+}} T_{ε}^{N, m} (v_{ε}) = - \infty$ , so that $T^{+} (v) = - \infty$ . Eventually, from the $L^{1} (T)$ -lower semicontinuity of $T^{+}$ , we deduce $N = N (v), m ⩾ m (v) \Rightarrow T^{+} (v) = - \infty . □$

8. Second-order estimate for $F_{ε}$

This section is devoted to prove estimate (1.5). In what follows, beside the hypotheses on W listed at the beginning of Section 2, we shall suppose also that there exists $δ \in (0, 1)$ so that $\{\begin{matrix} W (s) = \frac{α_{-}^{2}}{2} {(1 - s)}^{2}, & s \in (- 1 - δ, - 1 + δ), \\ W (s) = \frac{α_{+}^{2}}{2} {(1 - s)}^{2}, & s \in (1 - δ, 1 + δ) . \end{matrix}$ (8.1) Notice that, in this case, we have $β_{\pm} = 0 .$ (8.2)

Lemma 8.1 (Computation of

D_{+}^{'} (1)

We have $D_{+}^{'} (1) : = lim_{η \to 1^{-}} D_{+}^{'} (η) = 0 .$ (8.3)

Proof.

Using the additional assumption (8.1) on W, for $η \in (1 - δ, 1)$ we have for the function $D_{+}$ defined in (4.5), $D_{+} (η) = \int_{0}^{1 - δ} {\frac{1}{\sqrt{2 W (s) - 2 W (η)}} - \frac{1}{α_{+} \sqrt{{(1 - s)}^{2} - {(1 - η)}^{2}}}} d s,$ and therefore $D_{+}$ is of class $C^{\infty}$ in a left neighbourhood of $η = 1$ . Differentiating under the integral sign we get $D_{+}^{'} (η) = \int_{0}^{1 - δ} {\frac{W^{'} (η)}{{(2 W (s) - 2 W (η))}^{3 / 2}} + \frac{1 - η}{α_{+} {({(1 - s)}^{2} - {(1 - η)}^{2})}^{3 / 2}}} d s,$ and the assertion follows passing to the limit. □

Corollary 8.2 (Second-order expansion of

Q_{+}

and

L_{+}

We have $\begin{matrix} Q_{+} (η) = \frac{1}{α_{+}} log (\frac{2}{1 - η}) + c_{+} + o (1 - η), \\ L_{+} (η) = σ_{+} - \frac{α_{+}}{2} {(1 - η)}^{2} log (\frac{1}{1 - η}) \\ - \frac{α_{+}}{2} (log (2 κ_{+}) + \frac{1}{2}) {(1 - η)}^{2} + o ({(1 - η)}^{3}) \end{matrix}$ (8.4) as $η \to 1^{-}$ .

Proof.

The formula for $Q_{+}$ follows from (4.16), (4.8) and Lemma 8.1. The formula for $L_{+}$ follows by a direct computation as in the proof of Lemma 4.3, considering $lim_{η \to 1^{-}} \frac{L_{+} (η) - σ_{+} + (α_{+} / 2) {(1 - η)}^{2} log (1 / (1 - η)) + (α_{+} / 2) (log (2 κ_{+}) + 1 / 2) {(1 - η)}^{2}}{{(1 - η)}^{3}},$ applying de l’Hôpital’s theorem and using the expansion of $Q_{+}$ in (8.4) instead of (4.15) and (8.2). □

Theorem 8.3 (Second-order estimate from below).

Suppose that assumptions ( W1)–(W3) hold, and that in addition (8.1) holds. Let $(v_{ε})$ , v, k and $(d_{k}^{ε})$ be as in Theorem 6.1. Then $\begin{array}{rclr} F_{ε} (v_{ε}) & ⩾ & N (v) σ - α_{+} κ_{+}^{2} \sum_{k = 1}^{N (v)} e^{- α_{+} (d_{k}^{ε} / ε)} - α_{-} κ_{-}^{2} \sum_{k = 1}^{N (v)} e^{- α_{-} (d_{k}^{ε} / ε)} \\ + o (\sum_{k = 1}^{N (v)} e^{- (3 α_{+} / 2) (d_{k}^{ε} / ε)}) + o (\sum_{k = 1}^{N (v)} e^{- (3 α_{-} / 2) (d_{k}^{ε} / ε)}) & (8.5) \end{array}$ as $ε \to 0^{+}$ .

Proof.
Let $(v_{ε})$ , v, k and $d_{k}^{ε}$ be as in Theorem 6.1. We refine expansion (6.15): indeed, for $k \in S_{+} (v)$ , from (3.10), (6.13) and (8.4) it follows $1 - w_{ε} (\frac{d_{k}^{ε}}{2 ε}) = 2 κ_{+} e^{- α_{+} (d_{k}^{ε} / 2 ε) + o (1 - w_{ε} (d_{k}^{ε} / 2 ε))} = 2 κ_{+} e^{- α_{+} (d_{k}^{ε} / 2 ε) + o (e^{- α_{+} (d_{k}^{ε} / 2 ε)})} .$ (8.6)

Following the notation and the proof of Theorem 6.1 (see in particular (6.16)), we have to expand the quantity $L_{+} (w_{k}^{ε} (\frac{d_{k}^{ε}}{2 ε})) + \frac{d_{k}^{ε}}{2 ε} W (w_{k}^{ε} (\frac{d_{k}^{ε}}{2 ε})) .$ (8.7) Using (8.2), we can compute the coefficients of the terms of order $e^{- α_{+} (3 d_{k}^{ε} / 2 ε)}$ in (8.7), and show that they vanish, so that (8.5) follows. □

We conclude the paper with following result, the proof of which follows along the same lines of Theorem 8.3, in a much simpler way.
Theorem 8.4 (Second-order estimate from above).

Suppose that assumptions ( W1)–(W3) hold, and that in addition (8.1) holds. Let v and $(v_{ε})$ be as in Theorem 6.5. Then $\begin{array}{rclr} F_{ε} (v_{ε}) & ⩽ & N (v) σ - α_{+} κ_{+}^{2} \sum_{k = 1}^{N (v)} e^{- α_{+} (d_{k} / ε)} - α_{-} κ_{-}^{2} \sum_{k = 1}^{N (v)} e^{- α_{-} (d_{k} / ε)} \\ + o (\sum_{k = 1}^{N (v)} e^{- (3 α_{+} / 2) (d_{k} / ε)}) + o (\sum_{k = 1}^{N (v)} e^{- (3 α_{-} / 2) (d_{k} / ε)}) & (8.8) \end{array}$ as $ε \to 0^{+}$ .

Footnotes

Acknowledgements

The authors are grateful to the referee for several useful comments. The first and last authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA – INdAM).

References

[1]

Anzellotti and

Baldo, Asymptotic development by Γ-convergence, Appl. Math. Optimization 27 (1993), 105–123.

[2]

Anzellotti,

Baldo and

Orlandi, Γ-asymptotic development, the Cahn–Hilliard functional and curvatures, J. Math. Anal. Appl. 197 (1996), 908–924.

[3]

Bethuel,

Orlandi and

Smets, Slow motion for gradient systems with equal depth multiple-well potentials, J. Differ. Equations 250 (2011), 53–94.

[4]

Bongiorno,

L.E.

Scriven and

H.T.

Davis, Molecular theory of fluid interfaces, J. Colloid Interface Sci. 57 (1976), 462–475.

[5]

Braides and

Truskinovsky, Asymptotic expansion by Γ-convergence, Cont. Mech. Therm. 20 (2008), 21–62.

[6]

Braides and

C.I.

Zeppieri, Multiscale analysis of a prototypical model for the interaction between microstructure and surface energy, Interface Free Bound. 11 (2009), 61–118.

[7]

Cannarsa and

Sinestrari, Semiconcave Functions, Hamilton–Jacobi Equations, and Optimal Control, Progress in Nonlinear Differential Equations and Their Applications, Vol. 58, Birkhäuser, Boston, 2004.

[8]

Carr and

R.L.

Pego, Metastable patterns in solutions of

u_{t} = ε^{2} u_{x x} - f (u)

, Comm. Pure Appl. Math. 42 (1989), 523–576.

[9]

Dal Maso, An Introduction to Γ-Convergence, Birkhäuser, Boston, 1993.

10.

[10]

De Giorgi and

Franzoni, Su un tipo di convergenza variazionale, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 58 (1975), 842–850.

11.

[11]

Fusco and

J.K.

Hale, Slow motion manifolds, dormant instability, and singular perturbations, Dyn. Diff. Equation 1 (1989), 75–94.

12.

[12]

M.E.

Gurtin, On the two-phase Stefan problem with interfacial energy and entropy, Arch. Ration. Mech. Anal. 96 (1986), 199–241.

13.

[13]

M.E.

Gurtin, On phase transitions with bulk, interfacial, and boundary energy, Arch. Ration. Mech. Anal. 96 (1986), 243–264.

14.

[14]

Modica and

Mortola, Un esempio di

Γ^{-}

-convergenza, Boll. Un. Mat. Ital. B 14 (1977), 285–299.

15.

[15]

Sandier and

Serfaty, Gamma-convergence of gradient flows with applications to Ginzburg–Landau, Comm. Pure Appl. Math. 57 (2004), 1627–1672.

16.

[16]

Serfaty, Gamma-convergence of gradient flows on Hilbert and metric spaces and applications, Disc. Cont. Dyn. Systems A 31 (2011), 1427–1451.

Γ -type estimates for the one-dimensional Allen–Cahn’s action

Abstract

Keywords

1. Introduction

Lemma 4.1 (Expansion of D + ).

Lemma 5.1 (The functions z ε ).

Theorem 6.1 (First order estimate from below).

Footnotes

Acknowledgements

References

Lemma 4.1 (Expansion of $D_{+}$ ).

Lemma 5.1 (The functions $z_{ε}$ ).