A note on the one-dimensional critical points of the Ambrosio

Abstract

This note addresses the question of convergence of critical points of the Ambrosio–Tortorelli functional in the one-dimensional case under pure Dirichlet boundary conditions. An asymptotic analysis argument shows the convergence to two possible limits points: either a globally affine function or a step function with a single jump at the middle point of the space interval, which are both critical points of the one-dimensional Mumford–Shah functional under a Dirichlet boundary condition. As a byproduct, non minimizing critical points of the Ambrosio–Tortorelli functional satisfying the energy convergence assumption as in (Babadjian, Millot and Rodiac (2022)) are proved to exist.

Keywords

Mumford–Shah functional Ambrosio–Tortorelli functional Γ-convergence critical points brittle fracture

1. Introduction

This note can be seen as a companion to our paper [2] in which we address the convergence of critical points of the Ambrosio–Tortorelli functional. We refer to [2] and references therein for motivation on this topic related to image segmentation [6] or fracture mechanics [3]. We focus here on the one-dimensional case where the Ambrosio–Tortorelli functional is defined by $\begin{array}{rcl} {AT}_{ε} (u, v) & : = & \int_{0}^{L} (η_{ε} + v^{2}) {| u^{'} |}^{2} d x \\ (1.1) & + \int_{0}^{L} (ε {| v^{'} |}^{2} + \frac{{(v - 1)}^{2}}{4 ε}) d x for all (u, v) \in {[H^{1} (0, L)]}^{2}, \end{array}$ where $L > 0$ and $ε \to 0$ , $η_{ε} \to 0$ are infinitesimal parameters satisfying $0 < η_{ε} ≪ ε$ . This functional, originally introduced in [1] can be interpreted as a phase-field regularization of the Mumford–Shah functional (see [6]) $\begin{matrix} (1.2) & (u, v) \mapsto \{\begin{matrix} MS (u) : = \int_{0}^{L} | u^{'} |^{2} d x + # (J_{u}) & if \{\begin{matrix} u \in {SBV}^{2} (0, L), \\ v = 1 in (0, L), \end{matrix} \\ + \infty & otherwise . \end{matrix} \end{matrix}$ More precisely, it has been proved in [1] that ${AT}_{ε}$ Γ-converges in the ${[L^{2} (0, L)]}^{2}$ -topology as $ε \to 0$ to the Mumford–Shah functional. As a consequence, the fundamental theorem of Γ-convergence ensures the convergence of global minimizers $(u_{ε}, v_{ε})$ of ${AT}_{ε}$ (under suitable boundary conditions) to $(u, 1)$ as $ε \to 0$ where $u \in {SBV}^{2} (0, L)$ is a global minimizer of MS.

In the present work we address the asymptotic analysis of critical points of the Ambrosio–Tortorelli functional, i.e. is a solution $(u_{ε}, v_{ε})$ of the ordinary differential equation $\begin{matrix} (1.3) & \{\begin{matrix} {[(η_{ε} + v_{ε}^{2}) u_{ε}^{'}]}^{'} = 0 & in (0, L), \\ - ε v_{ε}^{″} + v_{ε} | u_{ε}^{'} |^{2} + \frac{v_{ε} - 1}{4 ε} = 0 & in (0, L), \\ u_{ε} (0) = 0, u_{ε} (L) = a, \\ v_{ε} (0) = v_{ε} (L) = 1 . \end{matrix} \end{matrix}$ If global minimizers always define critical points, the converse might fail due to the non convexity of ${AT}_{ε}$ . Note also that, in contrast with [4,5], we consider a pure Dirichlet problem. As in [4] there is a selection phenomenon of possible accumulation points of $(u_{ε}, v_{ε})$ .

According to [2, Remark 1.1], in this setting, a function $u \in {SBV}^{2} (0, L)$ is a critical point of the Mumford–Shah functional in ${SBV}^{2} (0, L)$ if and only if either $u (x) = \frac{a x}{L}$ for $x \in [0, L]$ or u is piecewise constant with a finite number of jumps ${\hat{J}}_{u} = {x_{1}, \dots, x_{m}}$ with $x_{i} \in [0, L]$ for all $i = 1, \dots, m$ . In the first case the energy of u is $\begin{matrix} MS (u) = \int_{0}^{L} {| \frac{a}{L} |}^{2} d x = \frac{a^{2}}{L}, \end{matrix}$ while in the second case $\begin{matrix} MS (u) = # ({\hat{J}}_{u} \cap [0, L]) = m . \end{matrix}$

Thus, if $a^{2} < L$ then $u (x) = a x / L$ is the global minimizer whereas if $a^{2} > L$ then a constant function with exactly one jump anywhere in the closed interval $[0, L]$ is a global minimizer (if $a^{2} = L$ then all previous functions are global minimizers). In particular, we have $\begin{matrix} (1.4) & min_{{SBV}^{2} (0, L)} MS = min {\frac{a^{2}}{L}, 1} . \end{matrix}$

We now consider a family of critical points of ${AT}_{ε}$ , i.e., a family ${(u_{ε}, v_{ε})}_{ε > 0}$ in ${[H^{1} (0, L)]}^{2}$ satisfying (1.3) together with the uniform energy bound $\begin{matrix} (1.5) & {AT}_{ε} (u_{ε}, v_{ε}) = \int_{0}^{L} (η_{ε} + v_{ε}^{2}) {| u_{ε}^{'} |}^{2} d x + \int_{0}^{L} (ε {| v_{ε}^{'} |}^{2} + \frac{{(v_{ε} - 1)}^{2}}{4 ε}) d x ⩽ C . \end{matrix}$

The following result extends [4, Theorem 2.2] to the case of Dirichlet boundary conditions for the phase-field variable v. It states that only two critical points of the Mumford–Shah functional are attainable through this asymptotic analysis procedure: either the affine solution (with no jumps) or the step function with a single jump at the middle point of the interval $(0, L)$ . It also improves [2, Theorem 1.2] in the one-dimensional case since the energy convergence assumption is no longer required.

Theorem 1.1.
Let ${(u_{ε}, v_{ε})}_{ε > 0} \subset {[H^{1} (0, L)]}^{2}$ be a family satisfying ( 1.3 ) and ( 1.5 ). Then, up to a subsequence $ε \to 0$ ,
$(u_{ε}, v_{ε}) \to (u_{}, 1)$ in ${[L^{2} (0, L)]}^{2}$ where* $u_{} \in {u_{jump}, u_{aff}}$ with* $\begin{matrix} (1.6) & u_{jump} = a 1_{[L / 2, L]}, u_{aff} (x) = \frac{a x}{L} for all x \in [0, L] . \end{matrix}$

$(η_{ε} + v_{ε}^{2}) | u_{ε}^{'} |^{2} L^{1} (0, L) \overset{}{⇀} | u_{}^{'} |^{2} L^{1} (0, L)$ weakly in* $M ([0, L])$ ,

$| ε | v_{ε}^{'} |^{2} - \frac{{(1 - v_{ε})}^{2}}{4 ε} \to 0$ strongly in $L^{1} (0, L)$ ,

$ε | v_{ε}^{'} |^{2} L^{1} (0, L) \overset{}{⇀} α δ_{\frac{L}{2}}$ weakly* in* $M ([0, L])$ , with $α = 0$ or $α = 1 / 2$ . Moreover, if $u_{} = u_{jump}$ , then* $α = 1 / 2$ .

Remark 1.1.
We emphasize that we must have $α = 1 / 2$ for $u_{} = u_{jump}$ . However, $α = 0$ does not necessarily implies that $u_{} = u_{aff}$ . Indeed $α = 1 / 2$ provided $v_{ε}$ has a v-jump at $L / 2$ in the terminology of [4], i.e., as soon as $v_{ε} (L / 2) ⩽ \sqrt{C_{} ε}$ for some constant $C_{} > 0$ . However it could happen that this v-jump disappears in the limit and does not create a discontinuity for $u_{}$ .

As in [4] we are in presence of a selection phenomenon since critical points of ${AT}_{ε}$ cannot approximate any critical points of MS but only specific ones. Here the selection phenomenon is much stronger than in [4] in the sense that only two critical points $u_{jump}$ and $u_{aff}$ of MS can be reached as limits of critical points of ${AT}_{ε}$ .

We also show the existence of a family of critical points of ${AT}_{ε}$ approximating $u_{jump}$ . Theorem 1.2.
There exists a family* ${(u_{ε}, v_{ε})}_{ε > 0} \subset {[H^{1} (0, L)]}^{2}$ satisfying ( 1.3 ) and ( 1.5 ) such that $\begin{matrix} (u_{ε}, v_{ε}) \to (u_{jump}, 1) strongly in {[L^{2} (0, L)]}^{2} as ε \to 0 . \end{matrix}$
Remark 1.2.
According to Theorem 1.1, we obtain that the family in Theorem 1.2 satisfies the energy convergence ${AT}_{ε} (u_{ε}, v_{ε}) \to MS (u_{jump})$ . Furthermore, if $a^{2} < L$ , then (1.4) ensures that $u_{jump}$ is not a global minimizer of MS. This shows the existence of non minimizing critical points of ${AT}_{ε}$ satisfying the assumption of convergence of energy in [2, Theorem 1.2].

2. Preliminary estimates

We start by using the first equation in (1.3) to find a constant $c_{ε} \in R$ such that $\begin{matrix} (2.1) & (η_{ε} + v_{ε}^{2}) u_{ε}^{'} = c_{ε}, \end{matrix}$ which implies that $u_{ε}^{'}$ has a constant sign. Since we assume $u_{ε} (0) = 0$ and $u_{ε} (L) = a > 0$ , we deduce that $u_{ε}^{'} ⩾ 0$ and $c_{ε} ⩾ 0$ . Then the second equation in (1.3) can be rewritten as $\begin{matrix} (2.2) & - ε v_{ε}^{″} + \frac{c_{ε}^{2} v_{ε}}{{(η_{ε} + v_{ε}^{2})}^{2}} + \frac{v_{ε} - 1}{4 ε} = 0 . \end{matrix}$

We observe that, thanks to the energy bound (1.5), $\begin{matrix} a c_{ε} = \int_{0}^{L} u_{ε}^{'} c_{ε} d x = \int_{0}^{L} (η_{ε} + v_{ε}^{2}) {| u_{ε}^{'} |}^{2} d x ⩽ {AT}_{ε} (u_{ε}, v_{ε}) ⩽ C, \end{matrix}$ hence ${c_{ε}}_{ε > 0}$ is bounded and, up to a subsequence, we can assume that $\begin{matrix} (2.3) & c_{ε} \to c_{0} . \end{matrix}$

As in [4, Lemma 3.2] (and using [2, Proposition 4.1]), we have the following result.

Lemma 2.1.
Let ${(u_{ε}, v_{ε})}_{ε > 0} \subset {[H^{1} (0, L)]}^{2}$ satisfying ( 1.3 ) and ( 1.5 ). Then, up to a subsequence, $(u_{ε}, v_{ε}) \to (u_{}, 1)$ strongly in* ${[L^{2} (0, L)]}^{2}$ with $u_{} \in {SBV}^{2} (0, L)$ . Furthermore,* $u_{ε}^{'} \to c_{0}$ a.e. in $(0, L)$ , $| D u_{} | ((0, L)) ⩽ a$ and* $c_{0} ⩽ a / L$ .

In the one-dimensional setting, the Noether type conservation law of [2, Proposition 4.2] reads as $\begin{matrix} {(\frac{{(1 - v_{ε})}^{2}}{4 ε} - ε {| v_{ε}^{'} |}^{2} - (η_{ε} + v_{ε}^{2}) {| u_{ε}^{'} |}^{2})}^{'} = 0 in (0, L), \end{matrix}$ and it implies the existence of a constant $d_{ε} \in R$ , sometimes called discrepancy, such that $\begin{matrix} (2.4) & \frac{{(1 - v_{ε})}^{2}}{4 ε} - ε {| v_{ε}^{'} |}^{2} - (η_{ε} + v_{ε}^{2}) {| u_{ε}^{'} |}^{2} = d_{ε} . \end{matrix}$ Thanks to the energy bound (1.5), it is easy to see that ${d_{ε}}_{ε > 0}$ is a bounded sequence, and thus (up to a further subsequence) $\begin{matrix} (2.5) & d_{ε} \to d_{0} . \end{matrix}$ It also ensures the following uniform bounds (see [4, Lemma 3.4]).
Lemma 2.2.
For $ε > 0$ small enough, $\begin{matrix} {‖ u_{ε}^{'} ‖}_{L^{\infty} (0, L)} ⩽ \frac{2}{\sqrt{ε η_{ε}}}, {‖ v_{ε}^{'} ‖}_{L^{\infty} (0, L)} ⩽ \frac{2}{ε} . \end{matrix}$ Moreover, if $c_{0} > 0$ , then the following refined estimates hold $\begin{matrix} {‖ u_{ε}^{'} ‖}_{L^{\infty} (0, L)} ⩽ \frac{2}{c_{0} ε}, min_{[0, L]} v_{ε} ⩾ \frac{c_{0} \sqrt{ε}}{2} . \end{matrix}$

We next show the following strong maximum principle.
Lemma 2.3.
Let $(u_{ε}, v_{ε}) \in {[H^{1} (0, L)]}^{2}$ satisfying ( 1.3 ), then $0 < v_{ε} < 1$ in $(0, L)$ .
Proof.
Let $x_{0}$ be a minimum point of $v_{ε}$ in $[0, L]$ . If $v_{ε} (x_{0}) = 1$ , using that $0 ⩽ v_{ε} ⩽ 1$ , we deduce that $v_{ε} \equiv 1$ in $[0, L]$ . Inserting into the second equation of (1.3), we find that $u_{ε}$ is a constant function in $[0, L]$ which is in contradiction with $u_{ε} (0) = 0$ and $u_{ε} (L) = a > 0$ . As a consequence of the Dirichlet boundary condition for $v_{ε}$ , we have $x_{0} \in (0, L)$ and thus $v_{ε}^{″} (x_{0}) ⩾ 0$ . If $v_{ε} (x_{0}) = 0$ , using again the second equation in (1.3) we find that $- ε v_{ε}^{″} (x_{0}) = \frac{1}{4 ε} > 0$ which is a contradiction. Therefore, $v_{ε} ⩾ v_{ε} (x_{0}) > 0$ in $(0, L)$ .

Likewise, let $x_{1}$ be a maximum point of $v_{ε}$ in $[0, L]$ . If $x_{1} \in (0, L)$ and $v_{ε} (x_{1}) = 1$ , then we use that $v_{ε}^{″} (x_{1}) ⩽ 0$ together with (2.2) to obtain that $c_{ε} = 0$ . This implies by (2.1) that $u_{ε}^{'} = 0$ which is a contradiction since $u_{ε} (0) = 0$ and $u_{ε} (L) = a > 0$ . It shows again that $v_{ε} < 1$ in $(0, L)$ . □

The selection phenomenon already observed in [4] is due to the following symmetry property which is similar to [4, Lemma 4.1].
Proposition 2.1.
Let ${(u_{ε}, v_{ε})}_{ε > 0}$ be a family in ${[H^{1} (0, L)]}^{2}$ satisfying ( 1.3 ) and ( 1.5 ). Then $v_{ε}$ possesses a unique critical point in $(0, L)$ located at $L / 2$ , which is a minimum of $v_{ε}$ on $[0, L]$ . Moreover, $v_{ε}$ is decreasing in $(0, L / 2)$ , increasing in $(L / 2, L)$ and the graph of $v_{ε}$ is symmetric with respect to the vertical line $x = L / 2$ .
Proof.
From Lemma 2.3, $v_{ε}$ cannot be identically constant equal to 1. Thus by Rolle’s theorem, $v_{ε}$ admits critical points in $(0, L)$ .

Let $x_{0} \in (0, L)$ be an arbitrary critical point of $v_{ε}$ in $(0, L)$ . If $x_{0} \in (0, L / 2)$ then the function $\begin{matrix} {\tilde{v}}_{ε} (x) = \{\begin{matrix} v_{ε} (x) & if x \in (0, x_{0}] \\ v_{ε} (2 x_{0} - x) & if x \in (x_{0}, 2 x_{0}) \end{matrix} \end{matrix}$ is a solution of (2.2) in the interval $(0, 2 x_{0})$ . In particular, $v_{ε}$ and ${\tilde{v}}_{ε}$ are two solutions of an ODE of the form $v_{ε}^{″} = f_{ε} (x, v_{ε})$ in $(x_{0}, 2 x_{0})$ for some function $f_{ε}$ of class $C^{2}$ with $v_{ε} (x_{0}) = {\tilde{v}}_{ε} (x_{0})$ and $v_{ε}^{'} (x_{0}) = {\tilde{v}}_{ε}^{'} (x_{0}) = 0$ . Cauchy–Lipschitz theorem yields in turn that $v_{ε} = {\tilde{v}}_{ε}$ in $(x_{0}, 2 x_{0})$ . In particular, $v_{ε} (2 x_{0}) = v_{ε} (0) = 1$ which contradicts Lemma 2.3 since $2 x_{0} \in (0, L)$ . Thus $x_{0} \in [L / 2, L)$ and a symmetric argument shows that $x_{0} \in (0, L / 2]$ . Finally, the only possibility left is $x_{0} = L / 2$ .

In particular, $v_{ε}$ admits a unique critical point in $(0, L)$ at the point $L / 2$ , which must be a minimum of $v_{ε}$ on $[0, L]$ . Moreover, the graph of $v_{ε}$ is symmetric with respect to the vertical line ${x = L / 2}$ . Since $v_{ε}$ is a smooth function satisfying $v_{ε} (0) = 1$ , $v_{ε} (L / 2) < 1$ and $v_{ε}^{'} \neq 0$ in $(0, L / 2)$ , we deduce that $v_{ε}$ is decreasing in $(0, L / 2)$ . By symmetry $v_{ε}$ is increasing in $(L / 2, L)$ . □

A crucial step in the proof of Theorem 1.1 is the following characterization of possible limiting slopes $c_{0}$ in (2.3), which strongly rests on the symmetry property of $v_{ε}$ . We refer to [4, Lemma 4.4] for the proof.
Lemma 2.4.
The limiting slope $c_{0}$ in ( 2.3 ) satisfies that either $c_{0} = 0$ or $c_{0} = a / L$ .

Using that $u_{ε}^{'} = \frac{c_{ε}}{η_{ε} + v_{ε}^{2}}$ in (2.4), we find that $\begin{matrix} \frac{{(1 - v_{ε})}^{2}}{4 ε} - ε {| v_{ε}^{'} |}^{2} - \frac{c_{ε}^{2}}{η_{ε} + v_{ε}^{2}} = d_{ε} . \end{matrix}$ Thus, since $v_{ε} ⩽ 1$ and $c_{ε}$ and $d_{ε}$ are bounded, we obtain that, for some constant $C_{} > 0$ independent of ε, $\begin{matrix} v_{ε}^{2} (L / 2) {(1 - v_{ε} (L / 2))}^{2} ⩽ (η_{ε} + v_{ε}^{2} (L / 2)) {(1 - v_{ε} (L / 2))}^{2} ⩽ C_{} ε . \end{matrix}$ This implies, thanks to the study of the function $X \mapsto X^{2} {(1 - X)}^{2}$ on $[0, 1]$ , that $\begin{matrix} either v_{ε} (\frac{L}{2}) ⩾ 1 - \sqrt{C_{} ε} or v_{ε} (\frac{L}{2}) ⩽ \sqrt{C_{} ε} . \end{matrix}$ In the latter case, $L / 2$ corresponds to a so-called v-jump according to the terminology of [4]. The previous dichotomy implies that either $v_{ε}$ converges uniformly to 1 or there exists exactly one v-jump which is a minimum of $v_{ε}$ located at $L / 2$ .
3. Proof of Theorem 1.1

We are now ready to prove items (i) and (ii) of Theorem 1.1, i.e., the selection principle for limit of critical points of ${AT}_{ε}$ and the convergence of the bulk energy.

Proof of (i) and (ii) in Theorem 1.1.
Step 1. Assume first that $v_{ε} (L / 2) ⩾ 1 - \sqrt{C_{} ε}$ . Then we have that $0 ⩽ 1 - v_{ε} ⩽ \sqrt{C_{} ε} \to 0$ uniformly in $[0, L]$ . For ε small enough we also have that $v_{ε} ⩾ 1 / 2$ so that the energy bound (1.5) yields $\begin{matrix} \frac{1}{2} \int_{0}^{L} {| u_{ε}^{'} |}^{2} d x ⩽ \int_{0}^{L} (η_{ε} + v_{ε}^{2}) {| u_{ε}^{'} |}^{2} d x ⩽ C . \end{matrix}$ Since ${u_{ε}}_{ε > 0}$ is bounded in $H^{1} (0, L)$ up to a subsequence we have that $u_{ε} ⇀ u_{}$ weakly in $H^{1} (0, L)$ with $u_{} \in H^{1} (0, L)$ . We can pass to the limit in the first equation of (2.1) using (2.3) to obtain that $u_{}^{'} = c_{0}$ in $(0, L)$ . Since $u_{} (0) = 0$ and $u_{} (L) = a$ we find that $u_{} = u_{aff}$ and $c_{0} = a / L$ . Moreover, thanks to the uniform convergence of $v_{ε}$ , $\begin{matrix} lim_{ε \to 0} \int_{0}^{L} {| u_{ε}^{'} |}^{2} d x = lim_{ε \to 0} \int_{0}^{L} (η_{ε} + v_{ε}^{2}) {| u_{ε}^{'} |}^{2} d x = lim_{ε \to 0} c_{ε} \int_{0}^{L} u_{ε}^{'} d x = c_{0} \int_{0}^{L} u_{aff}^{'} d x = \int_{0}^{L} {| u_{aff}^{'} |}^{2} d x . \end{matrix}$ It yields $u_{ε} \to u_{aff}$ strongly in $H^{1} (0, L)$ and, in particular $(η_{ε} + v_{ε}^{2}) | u_{ε}^{'} |^{2} \to | u_{aff}^{'} |^{2}$ strongly in $L^{1} (0, L)$ , hence also weakly* in $M ([0, L])$ .

Step 2. Assume now that $v_{ε} (\frac{L}{2}) < \sqrt{C_{} ε}$ . We first notice that $u_{ε} (L / 2) = a / 2$ . Indeed, thanks to the symmetry property of $v_{ε}$ and a change of variables, we find that $\begin{matrix} (3.1) & a = \int_{0}^{L} u_{ε}^{'} d x = \int_{0}^{L} \frac{c_{ε}}{η_{ε} + v_{ε}^{2}} d x = 2 \int_{0}^{L / 2} \frac{c_{ε}}{η_{ε} + v_{ε}^{2}} d x = 2 u_{ε} (\frac{L}{2}) . \end{matrix}$

We next claim that for each $0 < δ < L / 2$ , the function $v_{ε} \to 1$ uniformly on $[0, δ]$ . To this purpose, define $A_{ε} : = {x \in (0, L) : v_{ε} (x) ⩽ 1 - ε^{1 / 4}}$ . By the monotonicity properties of $v_{ε}$ , the set $A_{ε}$ is a closed interval centered in $L / 2$ . Thanks to the energy bound (1.5), $\begin{matrix} C ⩾ \int_{0}^{L} \frac{{(1 - v_{ε})}^{2}}{ε} d x ⩾ \frac{L^{1} (A_{ε})}{ε^{1 / 2}}, \end{matrix}$ which implies that $\begin{matrix} diam (A_{ε}) ⩽ C ε^{1 / 2} < \frac{L}{2} - δ \end{matrix}$ for ε small enough. Hence $A_{ε} \cap [0, δ] = \emptyset$ for ε small. In particular $v_{ε} \to 1$ uniformly on $[0, δ]$ , and then $u_{ε}^{'} = \frac{c_{ε}}{η_{ε} + v_{ε}^{2}} \to c_{0}$ uniformly on $[0, δ]$ . We deduce that $\begin{array}{l} u_{ε} (x) = \int_{0}^{x} u_{ε}^{'} (t) d t \to c_{0} x uniformly with respect to x \in [0, δ] . \end{array}$ Thus $u_{ε} (x) \to c_{0} x$ for a.e. $x \in (0, \frac{L}{2})$ , and we prove in the same way that $u_{ε} (x) \to a - c_{0} (L - x)$ for a.e. $x \in (\frac{L}{2}, L)$ . Since $c_{0} = 0$ or $c_{0} = a / L$ by Lemma 2.4, then we find that either $u_{} = u_{jump}$ or $u_{} = u_{aff}$ (see (1.6)). Observe that the case $u_{} = u_{jump}$ only occurs in the case $v_{ε} (L / 2) ⩽ \sqrt{C_{} ε}$ .

We finally show the convergence of the bulk energy. From the first equation in (1.3) we can write that $(η_{ε} + v_{ε}^{2}) | u_{ε}^{'} |^{2} = c_{ε} u_{ε}^{'}$ . Thus, for all $φ \in C_{c}^{\infty} (R)$ , $\begin{array}{rcl} \int_{0}^{L} (η_{ε} + v_{ε}^{2}) {| u_{ε}^{'} |}^{2} φ d x & = & c_{ε} \int_{0}^{L} u_{ε}^{'} φ d x \\ = & - c_{ε} \int_{0}^{L} u_{ε} φ^{'} d x + c_{ε} a φ (L) \to - c_{0} \int_{0}^{L} u_{} φ^{'} d x + c_{0} a φ (L) . \end{array}$ If $c_{0} = 0$ , then $u_{} = u_{jump}$ and $u_{jump}^{'} = 0$ . We thus get in that case, $\begin{matrix} - c_{0} \int_{0}^{L} u_{jump} φ^{'} d x + c_{0} a φ (L) = 0 = \int_{0}^{L} {| u_{jump}^{'} |}^{2} φ d x . \end{matrix}$ If $c_{0} = a / L$ , then $u_{} = u_{aff}$ and thus $\begin{matrix} - c_{0} \int_{0}^{L} u_{aff} φ^{'} d x + c_{0} a φ (L) = \frac{a}{L} \int_{0}^{L} u_{aff}^{'} φ d x = \int_{0}^{L} {| u_{aff}^{'} |}^{2} φ d x . \end{matrix}$ In any case, we obtain $\begin{matrix} \int_{0}^{L} (η_{ε} + v_{ε}^{2}) {| u_{ε}^{'} |}^{2} φ d x \to \int_{0}^{L} {| u_{}^{'} |}^{2} φ d x, \end{matrix}$ which proves the announced items (i) and (ii). □

From now on, the function u stands for either $u_{aff}$ or $u_{jump}$ . The argument in the previous proof actually shows that $u = u_{aff}$ if $c_{0} = a / L$ , while $u = u_{jump}$ if $c_{0} = 0$ .

At this stage, it remains to show the equipartition of energy and the convergence of the diffuse surface energy (points (iii) and (iv) in Theorem 1.1). The key argument is the following result stating that there is very few diffuse surface energy far way from $L / 2$ , the only possible limit jump point. The proof is an adaptation of [4, Lemma 6.1].
Lemma 3.1.
For every compact set* $K \subset [0, L] ∖ {L / 2}$ , there exists a constant $C_{K} > 0$ such that $\begin{matrix} \int_{K} (ε {| v_{ε}^{'} |}^{2} + \frac{{(1 - v_{ε})}^{2}}{4 ε}) d x ⩽ C_{K} ε^{1 / 4} . \end{matrix}$
Proof.
We already know that the set $A_{ε} : = {x \in (0, L) : v_{ε} (x) ⩽ 1 - ε^{1 / 4}}$ is a closed interval centred at $L / 2$ with $diam (A_{ε}) ⩽ C \sqrt{ε}$ . Let $δ < \frac{1}{2} dist (L / 2, K)$ . If $ε > 0$ is small enough then $A_{ε} \subset [L / 2 - δ, L / 2 + δ]$ , hence $K \cap A_{ε} = \emptyset$ . Since $K \subset V_{δ} : = [0, L] ∖ [L / 2 - δ, L / 2 + δ]$ , it suffices to show that $\begin{matrix} \int_{V_{δ}} (ε {| v_{ε}^{'} |}^{2} + \frac{{(1 - v_{ε})}^{2}}{4 ε}) d x ⩽ C_{K} ε^{1 / 4} . \end{matrix}$ We multiply the second equation in (1.3) by $v_{ε} - 1 \in H_{0}^{1} (0, L)$ and we integrate by parts to obtain $\begin{array}{rcl} \int_{V_{δ}} (ε {| v_{ε}^{'} |}^{2} + \frac{{(v_{ε} - 1)}^{2}}{4 ε}) d x & = & ε v_{ε}^{'} (L / 2 - δ) (v_{ε} (L / 2 - δ) - 1) \\ - ε v_{ε}^{'} (L / 2 + δ) (v_{ε} (L / 2 + δ) - 1) + \int_{V_{δ}} \frac{c_{ε}^{2} v_{ε} (1 - v_{ε})}{{(η_{ε} + v_{ε}^{2})}^{2}} d x . \end{array}$ By definition of $A_{ε}$ and $V_{δ}$ , we have $| 1 - v_{ε} | ⩽ ε^{1 / 4}$ on $V_{δ}$ . Using further the gradient bound for $v_{ε}$ in Lemma 2.2, we find that $\begin{matrix} \int_{V_{δ}} (ε {| v_{ε}^{'} |}^{2} + \frac{{(v_{ε} - 1)}^{2}}{4 ε}) d x ⩽ C_{K} ε^{1 / 4}, \end{matrix}$ which completes the proof of the lemma. □

Arguing as in [4, Lemma 6.3], we also have the following result which relates the limit slope $c_{0}$ to the limit $d_{0}$ (respectively defined in (2.3) and (2.5)).
Lemma 3.2.
The real numbers $c_{0}$ and $d_{0}$ satisfy $d_{0} + c_{0}^{2} = 0$ .

We are now in position to complete the proof of Theorem 1.1. Proof of (iii) and (iv) in Theorem 1.1.
Step 1. Let us consider the function $\begin{matrix} f_{ε} : = \frac{{(v_{ε} - 1)}^{2}}{4 ε} - ε {| v_{ε}^{'} |}^{2} = d_{ε} + (η_{ε} + v_{ε}^{2}) {| u_{ε}^{'} |}^{2} = d_{ε} + c_{ε} u_{ε}^{'} = d_{ε} + \frac{c_{ε}^{2}}{η_{ε} + v_{ε}^{2}} . \end{matrix}$

If $u_{} = u_{jump}$ , then $\begin{matrix} \int_{0}^{L} (η_{ε} + v_{ε}^{2}) {| u_{ε}^{'} |}^{2} d x \to 0, \end{matrix}$ hence $c_{ε} a = c_{ε} \int_{0}^{L} u_{ε}^{'} d x = \int_{0}^{L} (η_{ε} + v_{ε}^{2}) | u_{ε}^{'} |^{2} d x \to 0$ . It shows that $c_{0} = 0$ and thus $d_{0} = 0$ owing to Lemma 3.2. We thus infer that $\begin{matrix} \int_{0}^{L} | \frac{{(v_{ε} - 1)}^{2}}{4 ε} - ε {| v_{ε}^{'} |}^{2} | d x \to 0 . \end{matrix}$

Assume next that $u_{} = u_{aff}$ . In that case, we have $\begin{matrix} \int_{0}^{L} (η_{ε} + v_{ε}^{2}) {| u_{ε}^{'} |}^{2} d x ↛ 0, \end{matrix}$ and the same argument as before shows that $c_{0} \neq 0$ . Then $d_{0} \neq 0$ by Lemma 3.2. In particular $d_{ε} \neq 0$ for $ε > 0$ small enough. The function $f_{ε}$ reaches its maximum when $v_{ε}$ is minimal, i.e., at the point $L / 2$ . Since $L / 2$ is a critical point of $v_{ε}$ , we have $\begin{matrix} max_{[0, L]} f_{ε} = f_{ε} (L / 2) = \frac{{(v_{ε} (L / 2) - 1)}^{2}}{4 ε} ⩾ 0 . \end{matrix}$ Similarly, $f_{ε}$ reaches its minimum when $v_{ε}$ attains its maximum. Since $v_{ε}$ is maximal on the boundary with $v_{ε} (0) = v_{ε} (L) = 1$ we find that $\begin{matrix} min_{[0, L]} f_{ε} = f_{ε} (0) = - ε {| v_{ε}^{'} (0) |}^{2} ⩽ 0 . \end{matrix}$ As a consequence, there exists $s_{ε} \in (0, L / 2)$ such that $f_{ε} (s_{ε}) = 0$ . From Lemma 3.2, it follows that $\begin{matrix} v_{ε}^{2} (s_{ε}) = - η_{ε} - \frac{c_{ε}^{2}}{d_{ε}} \to - \frac{c_{0}^{2}}{d_{0}} = 1 . \end{matrix}$ Up to a subsequence, there exists $s_{0} \in [0, L / 2]$ such that $s_{ε} \to s_{0}$ . By monotonicity of $v_{ε}$ , we get that $v_{ε} 1_{[0, s_{ε}]} \to 1_{[0, s_{0}]}$ for a.e. $s \in (0, L / 2)$ . Hence using again Lemma 3.2, $\begin{matrix} (3.2) & d_{ε} s_{ε} + c_{ε} u_{ε} (s_{ε}) = \int_{0}^{s_{ε}} (d_{ε} + \frac{c_{ε}^{2}}{η_{ε} + v_{ε}^{2}}) d x \to \int_{0}^{s_{0}} (d_{0} + c_{0}^{2}) d x = 0 . \end{matrix}$ Using the symmetry of $v_{ε}$ , (3.1) and (3.2), we compute $\begin{array}{rcl} \int_{0}^{L} | \frac{{(v_{ε} - 1)}^{2}}{4 ε} - ε {| v_{ε}^{'} |}^{2} | d x & = & \int_{0}^{L} | f_{ε} | d x = 2 \int_{0}^{L / 2} | f_{ε} | d x \\ = & - 2 \int_{0}^{s_{ε}} f_{ε} d x + 2 \int_{s_{ε}}^{L / 2} f_{ε} d x \\ = & 2 d_{ε} (\frac{L}{2} - 2 s_{ε}) + 2 c_{ε} (u_{ε} (\frac{L}{2}) - 2 u_{ε} (s_{ε})) \\ \to & L d_{0} + c_{0} a = 0 . \end{array}$ It completes the proof of the equipartition of energy.

Step 2. We finally show the convergence of the diffuse surface energy. According to Lemma 3.1, we have $\begin{matrix} (ε {| v_{ε}^{'} |}^{2} + \frac{{(v_{ε} - 1)}^{2}}{4 ε}) L^{1} (0, L) \overset{}{⇀} μ weakly in M ([0, L]), \end{matrix}$ for some nonnegative measure $μ \in M ([0, L])$ supported on ${L / 2}$ , and thus of the form $μ = c δ_{L / 2}$ with $c ⩾ 0$ . On the one hand, since μ is concentrated at $L / 2$ , we have $\begin{matrix} c = μ ((0, L)) = lim_{ε \to 0} \int_{0}^{L} (ε {| v_{ε}^{'} |}^{2} + \frac{{(v_{ε} - 1)}^{2}}{4 ε}) d x . \end{matrix}$ On the other hand, the equipartition of energy ensures that $\begin{array}{l} \int_{0}^{L} (ε {| v_{ε}^{'} |}^{2} + \frac{{(v_{ε} - 1)}^{2}}{4 ε} - | v_{ε}^{'} | (1 - v_{ε})) d x & = \int_{0}^{L} {(\sqrt{ε} | v_{ε}^{'} | - \frac{1 - v_{ε}}{2 \sqrt{ε}})}^{2} d x \\ ⩽ \int_{0}^{L} | ε {| v_{ε}^{'} |}^{2} - \frac{| 1 - v_{ε} |^{2}}{4 ε} | d x \to 0 . \end{array}$ Since $v_{ε}^{'} ⩽ 0$ on $(0, L / 2)$ and $v_{ε}^{'} ⩾ 0$ on $(L / 2, L)$ , by symmetry of $v_{ε}$ with respect to the vertical axis ${x = L / 2}$ , we have $\begin{matrix} \int_{0}^{L} | v_{ε}^{'} | (1 - v_{ε}) d x = - 2 \int_{0}^{L / 2} v_{ε}^{'} (1 - v_{ε}) d x = {(1 - v_{ε} (L / 2))}^{2} . \end{matrix}$ If $v_{ε} (L / 2) ⩽ \sqrt{C_{} ε}$ , then $c = 1$ and $μ = δ_{L / 2}$ , while if $v_{ε} (L / 2) ⩾ 1 - \sqrt{C_{} ε}$ , then $c = 0$ and $μ = 0$ . Using again the equipartition of energy, we infer that $\begin{matrix} ε {| v_{ε}^{'} |}^{2} L^{1} (0, L) \overset{}{⇀} \frac{c}{2} δ_{L / 2} weakly in M ([0, L]), \end{matrix}$ so that the desired convergence holds with $α = \frac{c}{2} \in {0, \frac{1}{2}}$ . If $u_{} = u_{jump}$ , then we must have $v_{ε} (L / 2) ⩽ \sqrt{C_{} ε}$ , and it follows that $α = 1 / 2$ in that case. □

4. Proof of Theorem 1.2

This section is devoted to prove Theorem 1.2, following again arguments similar to those of [4, Section 5]. By the symmetry properties of Theorem 1.1 it suffices to construct a critical point $(u_{ε}, v_{ε})$ of ${AT}_{ε}$ in $(0, L / 2)$ such that $v_{ε} (0) = 1$ , $v_{ε}^{'} (L / 2) = 0$ , $u_{ε} (0) = 0$ and $u_{ε} (L / 2) = a / 2$ .

To this aim, let $α \in (0, 1)$ independent of ε, and set $\begin{array}{l} V : = {v \in H^{1} (0, L / 2) : v (0) = 1, v (\frac{L}{2}) ⩽ α}, \\ B : = {(u, v) \in H^{1} (0, L / 2) \times V : u (0) = 0, u (\frac{L}{2}) = \frac{a}{2}} . \end{array}$ For $(u, v) \in B$ and $0 ⩽ r ⩽ s ⩽ L / 2$ , we define the localized bulk and diffuse surface energies by $\begin{matrix} E_{ε} (u, v; r, s) = \int_{r}^{s} (η_{ε} + v^{2}) {| u^{'} |}^{2} d x, F_{ε} (v; r, s) : = \int_{r}^{s} (ε {| v^{'} |}^{2} + \frac{{(1 - v)}^{2}}{4 ε}) d x, \end{matrix}$ and the Ambrosio–Tortorelli energy localized on $(0, L / 2)$ by $\begin{matrix} {\tilde{AT}}_{ε} (u, v) : = E_{ε} (u, v; 0, L / 2) + F_{ε} (v; 0, L / 2) . \end{matrix}$

The following result has been established in [4, Section 5].

Lemma 4.1.

For all $x_{1}$ , $x_{2}$ and $x_{3} \in (0, L / 2)$ we have $\begin{matrix} F_{ε} (v; x_{1}, x_{3}) ⩾ | Φ (v (x_{1})) + Φ ((v (x_{3})) - 2 Φ (v (x_{2})) |, \end{matrix}$ with $Φ (t) = t - t^{2} / 2$ .

Using the monotone increasing character of Φ on $[0, 1]$ and choosing $x_{1} = x_{2} = 0$ and $x_{3} = L / 2$ , we get for all $v \in V$ $\begin{matrix} (4.1) & F_{ε} (v; 0, \frac{L}{2}) ⩾ Φ (1) - Φ (v (\frac{L}{2})) ⩾ \frac{1}{2} - Φ (α) . \end{matrix}$ Moreover, arguing as in [4, Section 5], we can show the existence of a minimizer $(u_{ε}, v_{ε})$ over $B$ of ${\tilde{AT}}_{ε}$ such that $\begin{matrix} (4.2) & \underset{ε \to 0}{lim sup} {\tilde{AT}}_{ε} (u_{ε}, v_{ε}) ⩽ \frac{1}{2} . \end{matrix}$

We will show that, for a convenient choice of $α \in (0, 1)$ , such a minimizer is a critical point of ${\tilde{AT}}_{ε}$ with the desired boundary conditions. The proof of the following result is similar to that of [4, Lemma 5.1 & Proposition 5.2].

Lemma 4.2.

There exists $α \in (0, 1)$ independent of $ε > 0$ such that if $(u_{ε}, v_{ε})$ is a minimizer over $B$ of ${\tilde{AT}}_{ε}$ , then it is a critical point of ${\tilde{AT}}_{ε}$ with $v_{ε}^{'} (L / 2) = 0$ for ε small enough.

Proof.

It is sufficient to show the existence of $α \in (0, 1)$ , independent of ε, such that if $(u_{ε}, v_{ε})$ is a minimizer over $B$ of ${\tilde{AT}}_{ε}$ , then $v_{ε} (L / 2) < α$ . Indeed, in that case the minimizer $(u_{ε}, v_{ε})$ belongs to the interior of $B$ and variations of the form $(u_{ε} + t ϕ, v_{ε} + t ψ)$ with $ϕ \in C_{c}^{\infty} ((0, L / 2))$ and $ψ \in C_{c}^{\infty} ((0, L / 2])$ are allowed. Let $α \in (0, 1)$ small enough so that $\begin{matrix} (4.3) & \frac{a^{2}}{L} - 2 Φ (α) > 0 . \end{matrix}$ Assume by contradiction that there exists $ε_{j} \to 0$ such that $v_{ε_{j}} (L / 2) = α$ for all $j \in N$ . Then consider the sequence $\begin{matrix} α_{j}^{*} : = min_{x \in [0, L / 2]} v_{ε_{j}} (x) ⩽ α . \end{matrix}$ Applying Lemma 4.1 with $x_{1} = 0$ , $x_{3} = L / 2$ and $x_{2} = y_{j}$ where $v_{ε_{j}} (y_{j}) = α_{j}^{*}$ leads to $\begin{array}{l} {\tilde{AT}}_{ε_{j}} (u_{ε_{j}}, v_{ε_{j}}) ⩾ \frac{1}{2} + Φ (α) - 2 Φ (α_{j}^{*}) . \end{array}$ We claim that for all $j \in N$ , $\begin{matrix} (4.4) & Φ (α) - 2 Φ (α_{j}^{*}) ⩾ \frac{Φ (α)}{2} > 0 . \end{matrix}$ Provided the claim is proved, we infer from (4.2) that $\begin{matrix} \frac{1}{2} ⩾ \underset{j \to \infty}{lim sup} {\tilde{AT}}_{ε_{j}} (u_{ε_{j}}, v_{ε_{j}}) ⩾ \frac{1}{2} + \frac{Φ (α)}{2}, \end{matrix}$ which is a contradiction since $Φ (α) > 0$ . We are now reduced to show (4.4).

Proof of the Claim. Let $α_{1} \in (0, α)$ be such that $Φ (α) ⩾ 4 Φ (α_{1})$ and assume by contradiction that $v_{ε_{j}} (x) ⩾ α_{1}$ for all $x \in [0, L / 2]$ . Using variations with compact support in $(0, L / 2)$ , we get that $(u_{ε_{j}}, v_{ε_{j}})$ solves $\begin{matrix} \{\begin{matrix} {(η_{ε_{j}} + v_{ε_{j}}^{2}) u_{ε_{j}}^{'})}^{'} = 0 & in (0, L / 2), \\ - ε_{j} v_{ε_{j}}^{″} + v_{ε_{j}} | u_{ε_{j}}^{'} |^{2} + \frac{v_{ε_{j}} - 1}{ε_{j}} = 0 & in (0, L / 2) . \end{matrix} \end{matrix}$ From the first equation we obtain that $u_{ε_{j}}^{'} (η_{ε_{j}} + v_{ε_{j}}^{2}) = c_{j}$ a.e. in $(0, L / 2)$ , for some constant $c_{j} \in R$ . The upper bound (4.2) shows that $v_{ε_{j}} \to 1$ in $L^{2} (0, L / 2)$ and $\begin{matrix} {\tilde{AT}}_{ε_{j}} (u_{ε_{j}}, v_{ε_{j}}) ⩾ \int_{0}^{L / 2} (η_{ε_{j}} + v_{ε_{j}}^{2}) {| u_{ε_{j}}^{'} |}^{2} d x = \int_{0}^{L / 2} c_{j} u_{ε_{j}}^{'} d x = \frac{a c_{j}}{2} . \end{matrix}$ It implies that ${c_{j}}_{j \in N}$ is bounded so that, up to a subsequence, $c_{j} \to c_{0}$ for some $c_{0} \in R$ . Since $v_{ε_{j}} ⩾ α_{1}$ , we deduce that ${u_{ε_{j}}^{'}}_{j \in N}$ is bounded in $L^{\infty} (0, L / 2)$ . Then Lebesgue’s dominated convergence yields $\begin{matrix} \frac{a}{2} = \int_{0}^{L / 2} u_{ε_{j}}^{'} d x = \int_{0}^{L / 2} \frac{c_{j}}{η_{ε_{j}} + v_{ε_{j}}^{2}} d x \to \int_{0}^{L / 2} c_{0} d x, \end{matrix}$ so that $c_{0} = a / L$ . Now, $\begin{matrix} \underset{j \to \infty}{lim inf} \int_{0}^{L / 2} (η_{ε_{j}} + v_{ε_{j}}^{2}) {| u_{ε_{j}}^{'} |}^{2} d x = \underset{j \to \infty}{lim inf} \int_{0}^{L / 2} c_{j} u_{ε_{j}}^{'} d x = c_{0} a = \frac{a^{2}}{L}, \end{matrix}$ and thus, by (4.1) and (4.2), $\begin{matrix} \frac{1}{2} ⩾ \underset{j \to \infty}{lim sup} {\tilde{AT}}_{ε_{j}} (u_{ε_{j}}, v_{ε_{j}}) ⩾ \frac{a^{2}}{L} + \frac{1}{2} - Φ (α), \end{matrix}$ which is in contradiction with (4.3).

We have thus proved by contradiction that ${min}_{[0, L / 2]} v_{ε_{j}} = α_{j}^{*} ⩽ α_{1}$ . Since we assumed that $Φ (α) ⩾ 4 Φ (α_{1}) ⩾ 4 Φ (α_{j}^{*})$ , we infer that (4.4) is satisfied. □

We can now conclude the proof of Theorem 1.2.

Proof of Theorem 1.2.

Let $(u_{ε}, v_{ε}) \in B$ and $α \in (0, 1)$ given by Lemma 4.2 (see (4.3)) so that $(u_{ε}, v_{ε})$ satisfies $\begin{matrix} \{\begin{matrix} {[(η_{ε} + v_{ε}^{2}) u_{ε}^{'}]}^{'} = 0 & in (0, L / 2), \\ - ε v_{ε}^{″} + v_{ε} | u_{ε}^{'} |^{2} + \frac{v_{ε} - 1}{4 ε} = 0 & in (0, L / 2), \\ u_{ε} (0) = 0, u_{ε} (L / 2) = a / 2, \\ v_{ε} (0) = 0, v_{ε}^{'} (L / 2) = 0 . \end{matrix} \end{matrix}$ By the first equation, there exists $c_{ε} \in R$ such that $u_{ε}^{'} = \frac{c_{ε}}{η_{ε} + v_{ε}^{2}}$ . Extending $v_{ε}$ to $(0, L)$ by symmetry with respect to the vertical axis ${x = L / 2}$ , we obtain a function (still denoted by $v_{ε}$ ) which belongs to $H^{1} (0, L)$ with $v_{ε} (0) = v_{ε} (L) = 1$ (this reflexion argument is possible since $v_{ε}^{'} (L / 2) = 0$ ). Note that the boundary conditions satisfied by $u_{ε}$ implies $\begin{matrix} c_{ε} = \frac{a}{2} {(\int_{0}^{L / 2} \frac{d x}{η_{ε} + v_{ε}^{2}})}^{- 1} = a {(\int_{0}^{L} \frac{d x}{η_{ε} + v_{ε}^{2}})}^{- 1}, \end{matrix}$ where the last equality holds because $v_{ε}$ is symmetric with respect to the vertical axis ${x = L / 2}$ . The function $u_{ε}$ is extended to $(0, L)$ (into a function still denoted by $u_{ε}$ ) by setting $\begin{matrix} u_{ε} (x) = \int_{0}^{x} \frac{c_{ε}}{η_{ε} + v_{ε}^{2} (t)} d t . \end{matrix}$ By construction, $(u_{ε}, v_{ε})$ solves $\begin{matrix} \{\begin{matrix} {[(η_{ε} + v_{ε}^{2}) u_{ε}^{'}]}^{'} = 0 & in (0, L), \\ - ε v_{ε}^{″} + v_{ε} | u_{ε}^{'} |^{2} + \frac{v_{ε} - 1}{4 ε} = 0 & in (0, L), \\ u_{ε} (0) = 0, u_{ε} (L) = a, \\ v_{ε} (0) = v_{ε} (L) = 0 . \end{matrix} \end{matrix}$ Moreover, the symmetry properties of $u_{ε}$ and $v_{ε}$ together with a change of variable yield $\begin{matrix} (4.5) & \underset{ε \to 0}{lim sup} {AT}_{ε} (u_{ε}, v_{ε}) = 2 \underset{ε \to 0}{lim sup} {\tilde{AT}}_{ε} (u_{ε}, v_{ε}) ⩽ 1 . \end{matrix}$ By Theorem 1.1, up to a subsequence, $(u_{ε}, v_{ε}) \to (u_{*}, 1)$ in ${[L^{2} (0, L)]}^{2}$ where $u_{*} \in {u_{jump}, u_{aff}}$ . Assume by contraction that $u_{*} = u_{aff}$ . According to [2, Proposition 4.1], we have $\begin{matrix} (4.6) & \frac{a^{2}}{L} ⩽ \underset{ε \to 0}{lim inf} \int_{0}^{L} (η_{ε} + v_{ε}^{2}) {| u_{ε}^{'} |}^{2} d x . \end{matrix}$ By (4.1) together with a change of variable and the symmetry property of $v_{ε}$ , we obtain $\begin{matrix} (4.7) & \underset{ε \to 0}{lim inf} \int_{0}^{L} (ε {| v_{ε}^{'} |}^{2} + \frac{{(1 - v_{ε})}^{2}}{4 ε}) d x = 2 \underset{ε \to 0}{lim inf} F_{ε} (v_{ε}; 0, \frac{L}{2}) ⩾ 1 - 2 Φ (α) . \end{matrix}$ Combining (4.5), (4.6) and (4.7) leads to $a^{2} / L - Φ (α) ⩽ 0$ , which is in contradiction with our choice of α in (4.3). Therefore $u_{*} = u_{jump}$ , and the proof is complete. □

References

Ambrosio and

V.M.

Tortorelli , On the approximation of free discontinuity problems, Boll. Un. Mat. Ital. B (7) 6(1) (1992), 105–123.

J.-F.

Babadjian ,

Millot and

Rodiac , On the convergence of critical points of the Ambrosio–Tortorelli functional, accepted in Ann. Inst. H. Poincaré C Anal. Non Linéaire.

Bourdin ,

G.A.

Francfort and

J.-J.

Marigo , The Variational Approach to Fracture, Springer, New York, 2008.

G.A.

Francfort ,

N.Q.

Le and

Serfaty , Critical points of Ambrosio–Tortorelli converge to critical points of Mumford–Shah in the one-dimensional Dirichlet case, ESAIM Control Optim. Calc. Var. 15(3) (2009), 576–598. doi:10.1051/cocv:2008041.

N.Q.

Le , Convergence results for critical points of the one-dimensional Ambrosio–Tortorelli functional with fidelity term, Adv. Differential Equations 15(3–4) (2010), 255–282.

Mumford and

Shah , Optimal approximations by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math. 42(5) (1989), 577–685. doi:10.1002/cpa.3160420503.

A note on the one-dimensional critical points of the Ambrosio–Tortorelli functional

Abstract

Keywords

1. Introduction

References