Locations of interior transition layers to inhomogeneous transition problems in higher -dimensional domains

Abstract

We consider the following inhomogeneous problems

\[ \begin{cases} \epsilon^{2}\mbox{div}(a(x)\nabla u(x))+f(x,u)=0 & \text{ in }\Omega,\\ \frac{\partial u}{\partial \nu}=0 & \text{ on }\partial \Omega,\\ \end{cases} \]

where $\Omega$ is a smooth and bounded domain in general dimensional space $\mathbb {R}^{N}$, $\epsilon >0$ is a small parameter and function $a$ is positive. We respectively obtain the locations of interior transition layers of the solutions of the above transition problems that are $L^{1}$-local minimizer and global minimizer of the associated energy functional.

1. Introduction

We study the following inhomogeneous transition problems

(1.1)\begin{equation} \begin{cases} \epsilon^{2}\mbox{div}(a(x)\nabla u(x))+f(x,u)=0 & \mbox{ in }~\Omega,\\ \frac{\partial u}{\partial \nu}=0 & \mbox{ on }~\partial \Omega,\\ \end{cases} \end{equation}

where $\Omega$ is a smooth and bounded domain in $\mathbb {R}^{N}$, $\nu$ is the outer unit normal to $\partial \Omega$, $\epsilon >0$ is a small parameter and function $a\in C^{1}(\bar {\Omega })$ is positive. The nonlinear term $f$ satisfies

• $(f_1)$ $f(x,\,\cdot )$ has two zeros $b_1(x),\, b_2(x)$ such that $b_1,\, b_2\in C^{1}(\Omega )$ and $b_1(x)< b_2(x)$ for all $x\in \bar {\Omega }$;

• $(f_2)$ $\partial _2f(x,\,b_1(x))<0$ and $\partial _2f(x,\,b_2(x))<0$ for all $x\in \bar {\Omega }$;

• $(f_3)$ For any given $x\in \bar {\Omega }$, $F(x,\,\cdot )\geq 0$. The function $\sqrt {a(\cdot )F(\cdot,\,\cdot )}$ is Lipschitz continuous. Here $F(x,\,u):=-\int ^{u}_{b_1(x)}f(x,\,\tau ){\rm d} \tau$.

A typical example of a function $f$ satisfying $(f_1)$–$(f_3)$ is

(1.2)\begin{equation} f(x,\tau)=V(x)\tilde{f}(\tau), \end{equation}

where $V$ is a strictly positive function and $\tilde {f}$ has precisely three zeros $\tilde {b}_1<0<\tilde {b}_2$, and $\int _{\tilde {b}_1}^{\tilde {b}_2}\tilde {f}(\tau ){\rm d} \tau =0$, moreover, ${\tilde {f}(\tau )}/{\tau }>\tilde {f}'(\tau )(\tau \neq 0).$

Another typical example of a function $f$ satisfying $(f_1)$–$(f_3)$ is

(1.3)\begin{equation} f(x,\tau)={-}(\tau-b_1(x))(\tau-b(x))(\tau-b_2(x)), \end{equation}

where $b_1(x)< b(x)< b_2(x)$ for all $x\in \Omega$.

The above two examples are related to inhomogeneous Allen–Cahn problem, which has its origin in the theory of phase transitions, see [5].

The corresponding energy functional of (1.1) is

\[ \bar{J}_\epsilon(u)=\displaystyle\int_{\Omega}\displaystyle\frac{\epsilon}{2}a(x)|\triangledown u|^{2}+ \frac{1}{\varepsilon} F(x,u) {\rm d}x. \]

For a smooth $(N-1)$-dimensional closed hypersurface $\Sigma$ contained in $\Omega$, we denote the domain enclosed by $\Sigma$ as $\Omega _{\Sigma }$. We denote by $\chi (A)$ the characteristic function related to set $A$.

Definition 1.1 A family $u_\epsilon$ of solutions to (1.1) is said to develop an interior transition layer, as $\epsilon \rightarrow 0$, with interface at some $(N-1)$-dimensional closed hypersurface $\Sigma _0\subset \Omega$ if

(1.4)\begin{equation} u_\epsilon\rightarrow u_0:=b_1\chi({\bar{\Omega}_{\Sigma_0}})+b_2\chi(\bar{\Omega}{\setminus} {\bar{\Omega}_{\Sigma_0}}) \text{ in } L^{1}(\Omega) \text{ as }\epsilon\rightarrow 0. \end{equation}

We introduce the set

\[ \Omega_-:=\left\{x\in\Omega: \displaystyle\int^{b_2(x)}_{b_1(x)}f(x,\tau){\rm d} \tau=0\right\}. \]

We call that $f$ satisfies the equal-area condition at the points in $\Omega _-$. Note that $\Omega _-=\left \{x\in \Omega : F(x,\,b_2(x))=0\right \}.$ It is well known that if $\Sigma _0$ is the interface of a family of solutions to (1.1) developing interior transition layer, then $\Sigma _0\subset \Omega _-$ (see [12]). Plainly, if $f$ is given by (1.2) then $\Omega _-=\Omega$. If $f$ is given by (1.3), we have $F(x,\,\tau )=\frac {1}{4}[(b_1(x)-b(x))^{2}-(\tau -b(x))^{2}]^{2}$ for those $x$ satisfying $2b(x)=b_1(x)+b_2(x)$, and so $\Omega _-=\{x\in \Omega : b(x)=\frac {1}{2}(b_1(x)+b_2(x))\}$. We denote $\Omega _+:=\Omega {\setminus} \bar {\Omega }_-$.

The following quantity plays an important role in determining location of interior layer

(1.5)\begin{equation} \Lambda(x):=\int^{b_2(x)}_{b_1(x)}\sqrt{a(x)F(x,\tau)}{\rm d} \tau. \end{equation}

We first recall some known results of transition layers of (1.1). For the case that $a(x)\equiv 1$ and $f$ is given by (1.2), in one-dimensional case, [25] shows that for an arbitrary subset of the local minimum points of $\Lambda (x)$, (1.1) admits a solution which has one layer near each point in the subset. Du and Gui [13] generalized the results of [25] to a two-dimensional case. Precisely, for a closed, non-degenerate geodesic $\Sigma _0$ relative to the integral $\int _\Sigma \Lambda$, (1.1) admits a solution whose layer locates near $\Sigma _0$. The corresponding results in general dimensional cases are established in [15, 16, 21, 32, 33]. For the corresponding fractional Laplacian, layer solutions are constructed in [14]. For $a(x)\equiv 1$ and $f$ given by (1.3), there are many known existence results of transition layer solutions, see [2–4, 6–12, 17, 18, 22, 31].

To construct layer solutions of a differential equation, the information of the location of the interface of a family of solutions is obviously very important and, in general, is not an easy task to find it.

In the homogeneous case, namely $a(x)\equiv 1$ and $f(x,\,u)\equiv f(u),$ classical theory of $\Gamma$-convergence developed in the 1970s and 1980s, showed a deep connection between this problem and the theory of minimal surfaces. By $\Gamma$-convergence theory, Modica [23] (see also [20, 24]) proved that a family $\{u_\epsilon \}$ of local minimizers of the energy functional with uniformly bounded energy must converge as $\epsilon \rightarrow 0$, up to subsequences, in $L^{1}$-sense to a function of the form $\chi _E-\chi _{E^{c}}$, where $\chi _E$ denotes characteristic function of a set $E$, and also that $\partial E$ has minimal perimeter.

For the inhomogeneous case, such as $a(x)\equiv 1$ and $f$ is given by (1.2), in one-dimensional case, transition layers of solutions to (1.1) can appear only near extremum points of $\Lambda (x)$ [26], and, in higher-dimensional cases, the authors in [21] establish a necessary condition for a closed hypersurface in $\Omega$ to support layers. For $a(x)\equiv 1$, $\Omega _-=\Omega$ and general $f$ satisfying assumptions $(f_1)$–$(f_3)$, in one-dimensional case, among other things, the authors in [27] proved the existence of solutions to (1.1) with interior transition layer and that the layer occurs only near some extremum point of $\Lambda (x)$.

Recently, in one-dimensional domain case ($\Omega =(0,\,1$)), for general $a(x)$ and $f$ satisfying conditions $(f_1)$–$(f_3)$, [28] obtains the following results.

Proposition 1.2 Suppose that a family $u_\epsilon$ of solutions to (1.1) develop an interior transition layer at $\bar {x}\in \mathcal {Q},$ where $\mathcal {Q}\subset \Omega _-$ is the connected component of $\Omega _-$ that $\bar {x}$ belongs to. Then

1. (i) if $u_\epsilon$ is a family of $L^{1}$-local minimizer of $\hat {J}_\epsilon,$ $\bar {x}$ is a local minimum point of $\Lambda (x)$ in $\mathcal {Q},$ where

   \[ \hat{J}_\epsilon(u):=\begin{cases} \bar{J}_\epsilon(u), & u\in H^{1}(0,1),\\ \infty, & u\in L^{1}(0,1){\setminus} H^{1}(0,1).\\ \end{cases} \]

2. (ii) if $u_\epsilon$ is a family of global minimizer of $\bar {J}_\epsilon,$ $\Lambda (\bar {x})=\min \{\Lambda (x): x\in \mathcal {Q}\}$.

What is the location of interior transition layer of minimizer ($L^{1}$-local or global) of the associated functional in general dimensional space? We will give a definite answer in this paper.

For small positive constant $\delta _0$, we define

\[ S:=\{x\in\Omega: \mbox{dist}(x,\Sigma_0)<2\delta_0\},\quad \Upsilon:=[{-}2\delta_0,2\delta_0]. \]

We parameterize elements $x\in S$ using their closest point $z$ in $\Sigma _0$ and their distance $t$ (with sign, negative in the dilation of $\Omega _{\Sigma _0}$). Precisely, we choose a system of coordinates $z$ on $\Sigma _0$, and denote by $\mathbf {n}(z)$ the unique unit normal vector to $\Sigma _0$ (at the point with coordinates $z$) pointing towards $\Omega {\setminus} \Omega _{\Sigma _0}$. Define the diffeomorphism $\Gamma : \Sigma _0\times \Upsilon \rightarrow S$ by

\[ \Gamma(z,t)=z+t\mathbf{n}(z). \]

We let the upper-case indices $I,\, J,\, \ldots$ run from $1$ to $N$, and the lower-case indices $i,\, j,\, \ldots$ run from $1$ to $N-1$. Using some local coordinates $(z_i)_{i=1,\ldots,N-1}$ on $\Sigma _0$, and letting $\varphi$ be the corresponding immersion into $\mathbb {R}^{N}$, we have

\[ \begin{cases} \displaystyle\frac{\partial \Gamma}{\partial z_i}(z,t)=\frac{\partial \varphi}{\partial z_i}(z)+t\kappa^{j}_i(z)\frac{\partial \varphi}{\partial z_j}(z) & \mbox{for}~ i=1, \ldots, N-1,\\ \displaystyle\frac{\partial \Gamma}{\partial t}(z,t)=\mathbf{n}(z),\\ \end{cases} \]

where $(\kappa ^{j}_i)$ are the coefficients of the mean-curvature operator on $\Sigma _0$. Let also $(\bar {g}_{ij})_{ij}$ be the coefficients of the metric on $\Sigma _0$ in the above coordinates $z$. Then, letting $g$ denote the metric on $\Omega$ induced by $\mathbb {R}^{N}$, we have

\[ g_{IJ}= \left( \begin{array}{ccccc} \{g_{ij}\} & 0\\ 0 & 1 \end{array} \right), \]

where

\begin{align*} g_{ij}& =\left(\frac{\partial \varphi}{\partial z_i}(z)+t\kappa^{m}_i(z)\frac{\partial \varphi}{\partial z_m}(z),\frac{\partial \varphi}{\partial z_j}(z)+t\kappa^{n}_j(z)\frac{\partial \varphi}{\partial z_n}(z)\right)\\ & =\bar{g}_{ij}+t(\kappa^{m}_i\bar{g}_{mj}+\kappa^{n}_j\bar{g}_{in})+t^{2}\kappa^{m}_i\kappa^{n}_j\bar{g}_{mn}. \end{align*}

We have, formally

\[ \det g=\det \bar{g} [1+t \mbox{Tr}(\bar{g}^{{-}1}\alpha)]+\mbox{o}(t), \]

where

\[ \alpha_{ij}=\kappa^{m}_i\bar{g}_{mj}+\kappa^{n}_j\bar{g}_{in}. \]

There holds

\[ (\bar{g}^{{-}1}\alpha)_{il}=\bar{g}^{lj}\alpha_{ij}=\bar{g}^{lj}(\kappa^{m}_i\bar{g}_{mj}+\kappa^{n}_j\bar{g}_{in}), \]

and hence

\[ \mbox{Tr}(\bar{g}^{{-}1}\alpha)=\bar{g}^{ij}(\kappa^{m}_i\bar{g}_{mj}+\kappa^{n}_j\bar{g}_{in})=2\bar{g}^{ij}\kappa^{m}_i\bar{g}_{mj}=2\kappa^{i}_i. \]

We recall that the quantity $\kappa ^{i}_i$ represents the mean curvature of $\Sigma _0$, we abbreviate $\kappa ^{i}_i$ as $\kappa$, and in particular it is independent of the choice of coordinates.

We have

\[ {\rm d}V_g=\sqrt{\det g}{\rm d}z{\rm d}t=(1+t\kappa+\mbox{o}(t))\sqrt{\det \bar{g}}{\rm d}z{\rm d}t=(1+t\kappa+\mbox{o}(t)){\rm d}V_{\bar{g}}{\rm d}t. \]

For $h$ satisfying $\|h\|_{L^{\infty }(\Sigma _0)}\leq 2\delta _0$, we define the perturbed closed $(N-1)$-dimensional hypersurface of $\Sigma _0$ as

\[ \Sigma_h:=\{\Gamma(z,h(z)): z\in\Sigma_0\}. \]

We also introduce

\[ J_\epsilon(u)=\begin{cases} \bar{J}_\epsilon(u), & u\in H^{1}(\Omega),\\ \infty, & u\in L^{1}(\Omega){\setminus} H^{1}(\Omega).\\ \end{cases} \]

We call that $u_\epsilon$ is a $L^{1}$-local minimizer of $J_\epsilon$ if there exists $\mu >0$ such that $J_\epsilon (u_\epsilon )\leq J_\epsilon (u)$ for any $u$ satisfying $\|u_\epsilon -u\|_{L^{1}(\Omega )}\leq \mu$. Each $L^{1}$-local minimizer of $J_\epsilon$ is a $H^{1}$-local minimizer of $\bar {J}_\epsilon$ as well, that is to say, it is a weak solution of (1.1). By the theory of regularity, it is a classical solution of (1.1).

Our main results are the followings.

Theorem 1.3 Suppose that a family $u_\epsilon$ of solutions to (1.1) develop an interior transition layer at $\Sigma _0\subset \mathcal {Q}$, where $\mathcal {Q}\subseteq \Omega _-$ is the connected component of $\Omega _-$ that $\Sigma _0$ belongs to. If $u_\epsilon$ is a family of $L^{1}$-local minimizer of $J_\epsilon$, then $\Sigma _0$ is a ‘local minimum’ surface of $\int _{\Sigma _h}\Lambda (x)$ in $\mathcal {Q}$ in the sense that there exists a $0<\sigma (\leq 2\delta _0)$ such that

\[ \displaystyle\int_{\Sigma_0}\Lambda=\min\left\{\int_{\Sigma_h}\Lambda: \|h\|_{L^{\infty}(\Sigma_0)}\leq\sigma ~\mbox{and}~ \|\nabla_{\bar{g}} h\|_{L^{\infty}(\Sigma_0)}=\mbox{o}(\epsilon^{{1}/{4}})\right\}. \]

Theorem 1.4 Besides the conditions of theorem 1.3, furthermore if $u_\epsilon$ is a family of global minimizer of $\bar {J}_\epsilon,$ then

\begin{align*} \int_{\Sigma_0}\Lambda& =\min\left\{\int_{\Sigma}\Lambda: \mbox{for any closed smooth }~(N-1){\text{-}dimensional} \right. \\ & \left.\mbox{nontrivial surface}~\Sigma\subset \mathcal{Q}~ \mbox{with} ~\Omega_+\backslash \Omega_{\Sigma}=\Omega_+\backslash \Omega_{\Sigma_0}\right\}. \end{align*}

Remark 1.5 If $\mathcal {Q}$ is a simply connected domain, then, for any closed $(N-1)$-dimensional surface $\Sigma \subset \mathcal {Q}$, both $\Omega _+\backslash \Omega _{\Sigma }$ and $\Omega _+\backslash \Omega _{\Sigma _0}$ are equal to $\Omega _+$, so the result of theorem 1.4 becomes

\[ \displaystyle\int_{\Sigma_0}\Lambda=\min\left\{\int_{\Sigma}\Lambda: \mbox{for any closed smooth nontrivial surface}~ \Sigma\subset \mathcal{Q}\right\}. \]

2. Preliminaries

We first recall definition of functions with bounded variation and a property to be used. The interested reader is referred to [19].

Definition 2.1 A function $\phi \in L^{1}(\Omega )$ is said to have bounded variation in $\Omega$ if

\begin{align*} \int_{\Omega}|D \phi|& :=\sup\left\{\int_{\Omega}\phi\,{\rm div}\,v{\rm d}x: v=(v_1,\ldots,v_N)\in C^{1}_0(\Omega,\mathbb{R}^{N}), \right.\\ & \quad\left. |v(x)|\leq 1~ \mbox{for}~x\in\Omega\right\}<\infty. \end{align*}

We define $BV(\Omega )$ as the space of all functions in $L^{1}(\Omega )$ with bounded variation. If $\phi \in BV(\Omega )$, then for any positive continuous function $v$, we have

(2.1)\begin{align} \int_{\Omega}v(x)|D \phi|& =\sup\left\{\int_{\Omega}\phi\,\mbox{div}\,w{\rm d}x: w=(w_1,\ldots,w_N)\in C^{1}_0(\Omega,\mathbb{R}^{N}), \right.\nonumber\\ & \quad\left. |w(x)|\leq v(x)~ \mbox{for}~x\in\Omega\right\}. \end{align}

Consider the following initial value problem

(2.2)\begin{equation} \begin{cases} \partial_\tau W(x,\tau)=\sqrt{\displaystyle\frac{2F(x,W(x,\tau))}{a(x)}},\\ W(x,0)=W_0(x),\\ \end{cases} \end{equation}

where $W_0\in C^{1}(\mathcal {Q})$ satisfies $b_1(x)\leq W_0(x)\leq b_2(x)$ for $x\in \mathcal {Q}$. This problem admits a unique solution $W(x,\,\tau )$ in $\mathcal {Q}\times \mathbb {R}$ and

\[ b_1(x)\leq W(x,\tau)\leq b_2(x),\quad \forall (x,\tau)\in\mathcal{Q}\times \mathbb{R}. \]

Moreover, $|\nabla _x W(x,\,\tau )|\in L^{\infty }(\mathcal {Q}\times \mathbb {R})$ and $\lim _{\tau \rightarrow -\infty }W(x,\,\tau )=b_1(x)$, $\lim _{\tau \rightarrow +\infty } W(x,\,\tau )=b_2(x)$. More precisely, there exists positive constants $q,\, \alpha$ depending on $F$ such that

$(W_1)$ for $\tau$ large enough, $|W(x,\,\tau )-b_2(x)|\leq q{\rm e}^{-\alpha \tau }$;

$(W_2)$ for $-\tau$ large enough, $|W(x,\,\tau )-b_1(x)|\leq q{\rm e}^{\alpha \tau }$.

The above properties of $W$ can be seen in [29] (see also [1, 30]).

3. Local minimum

We first establish a lower bound for $J_\epsilon ( u_\epsilon )$.

Lemma 3.1 Suppose that a family $u_\epsilon$ of solutions to (1.1) develop an interior transition layer at $\Sigma _0\subset \mathcal {Q}$, where $\mathcal {Q}\subseteq \Omega _-$ is the connected component of $\Omega _-$. If $u_\epsilon$ is a family of $L^{1}$-local minimizer of $J_\epsilon,$ then

(3.1)\begin{equation} J_\epsilon( u_\epsilon)\geq \sqrt{2}\int_{\Sigma_0}\Lambda {\rm d}V_{\bar{g}}+\int_{\Omega_+{{\setminus}} \Omega_{\Sigma_0}}\frac{1}{\epsilon}F(x,b_2(x)){\rm d}x+\mbox{o}(1), \end{equation}

where $\mbox {o}(1)$ means a quantity with limit $0$ as $\epsilon \rightarrow 0$.

Proof. First we assume that $\mathcal {Q}$ is simply connected. We have

(3.2)\begin{align} J_\epsilon(u_\epsilon)& =J_\epsilon(u_\epsilon,\Omega_+)+J_\epsilon(u_\epsilon,\Omega_-)\geq J_\epsilon(u_\epsilon,\Omega_+)+J_\epsilon(u_\epsilon,\mathcal{Q})\\ & \geq \int_{\Omega_+}\frac{1}{\epsilon}F(x,b_2(x)){\rm d}x+J_\epsilon(u_\epsilon,\mathcal{Q}). \nonumber \end{align}

Set

\[ \mathbb{U}:=\{v\in C_0^{1}(\mathcal{Q},\mathbb{R}^{N}): |v(x)|\leq 1\}, \]

then we have

\begin{align*} J_\epsilon(u_\epsilon,\mathcal{Q})& =\int_{\mathcal{Q}}\frac{\epsilon}{2}a(x)|\triangledown u_\epsilon|^{2}+ \frac{1}{\epsilon} F(x,u_\epsilon) {\rm d}x\\ & \geq \sqrt{2}\int_{\mathcal{Q}}|\nabla u_\epsilon|\sqrt{a(x)F(x,u_\epsilon)}\\ & =\sup_{v\in\mathbb{U}}\left\{\sqrt{2}\int_{\mathcal{Q}}\nabla u_\epsilon\cdot v\sqrt{a(x)F(x,u_\epsilon)}\right\}. \end{align*}

If we denote

\[ \psi_\epsilon(x):=\displaystyle\int_{b_1(x)}^{u_\epsilon(x)}\sqrt{a(x)F(x,\tau)}{\rm d} \tau, \]

then

\begin{align*} & J_\epsilon(u_\epsilon,\mathcal{Q}) \\ & \quad\geq \sup_{v\in\mathbb{U}}\left\{\sqrt{2}\int_{\mathcal{Q}}\left[\nabla \psi_\epsilon\cdot v-\int_{b_1(x)}^{u_\epsilon(x)}\nabla\left(\sqrt{a(x)F(x,\tau)}\right)\cdot v{\rm d} \tau\right]{\rm d}x\right\}\\ & \quad=\sup_{v\in\mathbb{U}}\left\{-\sqrt{2}\int_{\mathcal{Q}}\int_{b_1(x)}^{u_\epsilon(x)}\left[\sqrt{a(x)F(x,\tau)}\, \mbox{div}\,v \right.\right.\\ & \qquad\left.\left. +\nabla\left(\sqrt{a(x)F(x,\tau)}\right)\cdot v\right]{\rm d} \tau {\rm d}x\right\}. \end{align*}

Combining the limit $u_\epsilon \rightarrow u_0$ in $L^{1}(\Omega )$ and the $L^{\infty }$ boundedness of the several quantities $\sqrt {a(x)F(x,\,\tau )},\, \nabla (\sqrt {a(x)F(x,\,\tau )}),\, v,\, \mbox {div}\,v$, we have

(3.3)\begin{align} & \lim_{\epsilon\rightarrow 0}J_\epsilon(u_\epsilon,\mathcal{Q})\nonumber\\ & \quad\geq\sup_{v\in\mathbb{U}}\left\{-\sqrt{2}\int_{\mathcal{Q}}\int_{b_1(x)}^{u_0(x)} [\sqrt{a(x)F(x,\tau)} \,\mbox{div}\,v \right.\nonumber\\ & \qquad\left. +\nabla\left(\sqrt{a(x)F(x,\tau)}\right)\cdot v]{\rm d} \tau {\rm d}x\right\}\nonumber\\ & \quad=\sup_{v\in\mathbb{U}}\left\{-\sqrt{2}\int_{\mathcal{Q}}\chi(u_0(x)=b_2(x)) \right.\nonumber\\ & \qquad\left.\times\int_{b_1(x)}^{b_2(x)}\left[\sqrt{a(x)F(x,\tau)} \,\mbox{div}\,v+\nabla\left(\sqrt{a(x)F(x,\tau)}\right)\cdot v\right]{\rm d} \tau {\rm d}x\right\} \nonumber\\ & \quad=\sup_{v\in\mathbb{U}}\left\{-\sqrt{2}\int_{\mathcal{Q}}\chi(u_0(x)=b_2(x)) \right.\nonumber\\ & \qquad\left.\times\mbox{div}\left[\int_{b_1(x)}^{b_2(x)}\sqrt{a(x)F(x,\tau)} v {\rm d} \tau\right] {\rm d}x\right\}\nonumber\\ & \quad=\sqrt{2}\int_{\mathcal{Q}}\int_{b_1(x)}^{b_2(x)}|\nabla\chi(u_0(x)=b_2(x))|\sqrt{a(x)F(x,\tau)} {\rm d} \tau {\rm d}x\nonumber\\ & \quad=\sqrt{2}\int_{\Sigma_0}\int_{b_1(x)}^{b_2(x)}\sqrt{a(x)F(x,\tau)} {\rm d} \tau {\rm d}V_{\bar{g}}\nonumber\\ & \quad=\sqrt{2}\int_{\Sigma_0}\Lambda(x) {\rm d}V_{\bar{g}}. \end{align}

Note that $\Omega _+{\setminus} \Omega _{\Sigma _0}=\Omega _+$, since $\mathcal {Q}$ is a simply connected domain. From this and (3.2), (3.3), we obtain (3.1).

For the case that $\mathcal {Q}$ is multiply connected, (3.2) becomes

\[ J_\epsilon(u_\epsilon)\geq \displaystyle\int_{\Omega_+{{\setminus}} \Omega_{\Sigma_0}}\displaystyle\frac{1}{\epsilon}F(x,b_2(x)){\rm d}x+J_\epsilon(u_\epsilon,\mathcal{Q}), \]

and the same argument as that of the simply connected domain case gives the desired inequality (3.1).

We further establish an upper bound for $J_\epsilon ( u_\epsilon )$.

Lemma 3.2 Under the conditions of lemma 3.1, then for any $h$ satisfying $\|h\|_{L^{\infty }(\Sigma _0)}\leq \sigma$ for some $\sigma \leq 2\delta _0$ and $\|\nabla _{\bar {g}} h\|_{L^{\infty }(\Sigma _0)}=\mbox {o}(\epsilon ^{{1}/{4}})$, we have

(3.4)\begin{equation} J_\epsilon( u_\epsilon)\leq \sqrt{2}\int_{\Sigma_h}\Lambda {\rm d}V_{\bar{g}}+\int_{\Omega_+{{\setminus}} \Omega_{\Sigma_0}}\frac{1}{\epsilon}F(x,b_2(x)){\rm d}x+\mbox{o}(1). \end{equation}

Proof. First we also assume that $\mathcal {Q}$ is simply connected. We borrow the idea of [30] (see also [28]) to define a sequence of functions $b_\epsilon (z,\,t;\tau ): \Sigma _0\times \Upsilon \times \Upsilon \rightarrow \mathbb {R}$

\[ b_\epsilon(z,t;\tau)= \begin{cases} b_2(z,t), & 2\sqrt{\epsilon}\leq\tau<2\delta_0,\\ [b_2(z,t)-W(z,t;1/\sqrt{\epsilon})]\displaystyle\frac{\tau-2\sqrt{\epsilon}}{\sqrt{\epsilon}}+b_2(z,t), & \sqrt{\epsilon}<\tau<2\sqrt{\epsilon},\\ W(z,t;\tau/\epsilon), & |\tau|\leq\sqrt{\epsilon},\\ [W(z,t;-1/\sqrt{\epsilon})-b_1(z,t)]\displaystyle\frac{\tau+2\sqrt{\epsilon}}{\sqrt{\epsilon}}+b_1(z,t), & -2\sqrt{\epsilon}<\tau<{-}\sqrt{\epsilon},\\ b_1(z,t), & -2\delta_0<\tau\leq{-}2\sqrt{\epsilon},\\ \end{cases} \]

where $W$ is the solution of (2.2). Given $h$ satisfying $\|h\|_{L^{\infty }(\Sigma _0)}\leq \sigma$ and $\|\nabla _{\bar {g}} h\|_{L^{\infty }(\Sigma _0)}=\mbox {o}(\epsilon ^{{1}/{4}})$, we define $\rho _\epsilon : \Omega \rightarrow \mathbb {R}$ by

\[ \rho_\epsilon(x)= \begin{cases} b_2(x), & x\in\Omega{\setminus}\Omega_{\Sigma_{\delta_0}},\\ b_\epsilon(z,t;t-h(z)), & x=\varphi(z)+t\mathbf{n}(z)\in\Omega_{\Sigma_{\delta_0}}{\setminus}\Omega_{\Sigma_{-\delta_0}},\\ b_1(x), & x\in\Omega_{\Sigma_{-\delta_0}}.\\ \end{cases} \]

Claim: For any given $\mu >0$, there exist $\epsilon _0(\mu )>0$ and $\sigma (\mu )>0$, such that for all $\epsilon <\epsilon _0$ we have $\|u_\epsilon -\rho _\epsilon \|_{L^{1}(\Omega )}\leq \mu$.

Indeed, if we introduce

\[ \rho_0:=b_1\chi({\bar{\Omega}_{\Sigma_h}})+b_2\chi(\bar{\Omega}{\setminus} {\bar{\Omega}_{\Sigma_h}}), \]

then we know that there exists $\sigma (\mu )$ less than $2\delta _0$ such that for $\|h\|_{L^{\infty }(\Sigma _0)}\leq \sigma (\mu )$, the following inequality holds $\|u_0-\rho _0\|_{L^{1}(\Omega )}<\frac {\mu }{2}$. Hence, to prove the claim it is only need to show that $\rho _\epsilon \rightarrow \rho _0$ in $L^{1}(\Omega )$ as $\epsilon \rightarrow 0$.

By the definitions of $\rho _\epsilon$ and $\rho _0$, we have that

\begin{align*} & \int_{\Omega}|\rho_\epsilon(x)-\rho_0(x)|{\rm d}x =\int_{\Omega_{\Sigma_{h+2\sqrt{\epsilon}}}{\setminus}\Omega_{\Sigma_{h-2\sqrt{\epsilon}}}}|\rho_\epsilon(x)-\rho_0(x)|{\rm d}x\\ & \quad=\int_{\Omega_{\Sigma_{h+2\sqrt{\epsilon}}}{\setminus}\Omega_{\Sigma_{h+\sqrt{\epsilon}}}}|\rho_\epsilon(x)-\rho_0(x)|+ \int_{\Omega_{\Sigma_{h-\sqrt{\epsilon}}}{\setminus}\Omega_{\Sigma_{h-2\sqrt{\epsilon}}}}|\rho_\epsilon(x)-\rho_0(x)|\\ & \qquad+ \int_{\Omega_{\Sigma_{h+\sqrt{\epsilon}}}{\setminus}\Omega_{\Sigma_{h-\sqrt{\epsilon}}}}|\rho_\epsilon(x)-\rho_0(x)|{\rm d}x. \end{align*}

For the first integral of the right hand side we have

\begin{align*} & \int_{\Omega_{\Sigma_{h+2\sqrt{\epsilon}}}{\setminus}\Omega_{\Sigma_{h+\sqrt{\epsilon}}}}|\rho_\epsilon(x)-\rho_0(x)|{\rm d}x \\ & \quad=\frac{1}{\sqrt{\epsilon}}\int_{\sqrt{\epsilon}}^{2\sqrt{\epsilon}}\int_{\Sigma_0}[b_2(z,h(z)+\mu)-W(z,h(z)+\mu;1/\sqrt{\epsilon})] \\ & \qquad\times |\mu-2\sqrt{\epsilon}| [1+(h(z)+\mu)\kappa+\mbox{o}(h(z)+\mu)]{\rm d}V_{\bar{g}}{\rm d}\mu \\ & \quad\leq \frac{C}{\sqrt{\epsilon}}\int_{\sqrt{\epsilon}}^{2\sqrt{\epsilon}}(2\sqrt{\epsilon}-\mu){\rm d}\mu \\ & \quad=\mbox{O}(\sqrt{\epsilon}). \end{align*}

Analogously,

\[ \displaystyle\int_{\Omega_{\Sigma_{h-\sqrt{\epsilon}}}{\setminus}\Omega_{\Sigma_{h-2\sqrt{\epsilon}}}}|\rho_\epsilon(x)-\rho_0(x)|{\rm d}x=\mbox{O}(\sqrt{\epsilon}). \]

We have

\begin{align*} & \int_{\Omega_{\Sigma_{h+\sqrt{\epsilon}}}{\setminus}\Omega_{\Sigma_{h-\sqrt{\epsilon}}}}|\rho_\epsilon(x)-\rho_0(x)|{\rm d}x \\ & \quad=\int_{\Omega_{\Sigma_{h+\sqrt{\epsilon}}}{\setminus}\Omega_{\Sigma_{h}}}|\rho_\epsilon(x)-\rho_0(x)|{\rm d}x+ \int_{\Omega_{\Sigma_{h}}{\setminus}\Omega_{\Sigma_{h-\sqrt{\epsilon}}}}|\rho_\epsilon(x)-\rho_0(x)|{\rm d}x\\ & \quad=\int_{0}^{\sqrt{\epsilon}}\int_{\Sigma_{h+\mu}}|b_2(z,h(z)+\mu)-W(z,h(z)+\mu;\mu/\epsilon)|{\rm d}V_{\bar{g}}{\rm d}\mu \\ & \qquad+\int_{-\sqrt{\epsilon}}^{0}\int_{\Sigma_{h+\mu}}|b_1(z,h(z)+\mu)-W(z,h(z)+\mu;\mu/\epsilon)|{\rm d}V_{\bar{g}}{\rm d}\mu \\ & \quad=\mbox{O}(\sqrt{\epsilon}). \end{align*}

All in all we obtain that $\rho _\epsilon \rightarrow \rho _0$ in $L^{1}(\Omega )$ as $\epsilon \rightarrow 0$.

We decompose

\begin{align*} J_\epsilon(\rho_\epsilon)& =J_\epsilon(\rho_\epsilon,\Omega{\setminus}\Omega_{\Sigma_{h+2\sqrt{\epsilon}}}) +J_\epsilon(\rho_\epsilon,\Omega_{\Sigma_{h+2\sqrt{\epsilon}}}{\setminus}\Omega_{\Sigma_{h+\sqrt{\epsilon}}}) \\ & \quad+J_\epsilon(\rho_\epsilon,\Omega_{\Sigma_{h+\sqrt{\epsilon}}}{\setminus}\Omega_{\Sigma_{h-\sqrt{\epsilon}}}) +J_\epsilon(\rho_\epsilon,\Omega_{\Sigma_{h-\sqrt{\epsilon}}}{\setminus}\Omega_{\Sigma_{h-2\sqrt{\epsilon}}}) \\ & \quad+J_\epsilon(\rho_\epsilon,\Omega_{\Sigma_{h-2\sqrt{\epsilon}}}). \end{align*}

From the definition of $\rho _\epsilon$, we have

(3.5)\begin{align} J_\epsilon(\rho_\epsilon,\Omega{\setminus}\Omega_{\Sigma_{h+2\sqrt{\epsilon}}})& =\int_{\Omega{\setminus}\Omega_{\Sigma_{h+2\sqrt{\epsilon}}}}\frac{\epsilon}{2}a(x)|\nabla b_2|^{2}+\frac{1}{\epsilon}F(x,b_2(x)){\rm d}x \nonumber\\ & =\frac{1}{\epsilon}\int_{\Omega{\setminus}\Omega_{\Sigma_{h+2\sqrt{\epsilon}}}}F(x,b_2(x)){\rm d}x+\mbox{O}(\epsilon) \nonumber\\ & =\frac{1}{\epsilon}\int_{\Omega_+}F(x,b_2(x)){\rm d}x+\mbox{O}(\epsilon), \end{align}

where in the last equality we used the facts that $F(x,\,b_2(x))=0$ in $\Omega _-$, and $\Omega _+{\setminus} \Omega _{\Sigma _{h+2\sqrt {\epsilon }}}=\Omega _+$ in virtue of the simply connectedness of $\mathcal {Q}$.

Similarly, we have

(3.6)\begin{equation} J_\epsilon(\rho_\epsilon,\Omega_{\Sigma_{h-2\sqrt{\epsilon}}})=\mbox{O}(\epsilon). \end{equation}

We have

\[ J_\epsilon(\rho_\epsilon,\Omega_{\Sigma_{h+2\sqrt{\epsilon}}}{\setminus}\Omega_{\Sigma_{h+\sqrt{\epsilon}}})= \displaystyle\frac{1}{\epsilon}\displaystyle\int_{\Omega_{\Sigma_{h+2\sqrt{\epsilon}}}{\setminus}\Omega_{\Sigma_{h+\sqrt{\epsilon}}}}F(x,\rho_\epsilon(x)){\rm d}x+\mbox{o}(\epsilon). \]

Recalling that $F(x,\,b_2(x))=0=F_u(x,\,b_2(x))$ and $F_{uu}(x,\,b_2(x))>0$, we have

\begin{align*} & \frac{1}{\epsilon}\int_{\Omega_{\Sigma_{h+2\sqrt{\epsilon}}}{\setminus}\Omega_{\Sigma_{h+\sqrt{\epsilon}}}}F(x,\rho_\epsilon(x)){\rm d}x\\ & \quad=\frac{1}{\epsilon}\int_{\sqrt{\epsilon}}^{2\sqrt{\epsilon}}\int_{\Sigma_{h+\mu}} F\left(z,h(z)+\mu,\left[b_2(z,h+\mu)-W\left(z,h+\mu;\frac{1}{\sqrt{\epsilon}}\right)\right] \right.\nonumber\\ & \qquad\left. \times\frac{\mu-2\sqrt{\epsilon}}{\sqrt{\epsilon}}+b_2(z,h+\mu)\right){\rm d}V_{\bar{g}}{\rm d}\mu \\ & \quad\leq \frac{1}{\epsilon}\int_{\sqrt{\epsilon}}^{2\sqrt{\epsilon}}\int_{\Sigma_{h+\mu}} F\left(z,h+\mu,q{\rm e}^{-\alpha/\sqrt{\epsilon}}+b_2(z,h+\mu)\right){\rm d}V_{\bar{g}}{\rm d}\mu \\ & \quad= \frac{1}{\epsilon}\int_{\sqrt{\epsilon}}^{2\sqrt{\epsilon}}\int_{\Sigma_{h+\mu}} \left[F\left(z,h+\mu,q{\rm e}^{-\alpha/\sqrt{\epsilon}}+b_2(z,h+\mu)\right) \right.\\ & \qquad\left. -F(z,h+\mu,b_2(z,h+\mu))\right]{\rm d}V_{\bar{g}}{\rm d}\mu \\ & \quad\leq\frac{1}{\epsilon}\int_{\sqrt{\epsilon}}^{2\sqrt{\epsilon}}{\rm e}^{-\alpha/\sqrt{\epsilon}}\gamma_\epsilon {\rm d}\mu \\ & \quad=\frac{\sqrt{\epsilon}}{\epsilon}{\rm e}^{-\alpha/\sqrt{\epsilon}}\gamma_\epsilon, \end{align*}

where

\begin{align*} \gamma_\epsilon& :=q_1\sup\left\{\int_{\Sigma_{t}}F_u(z,t,\tau){\rm d}V_{\bar{g}}:\right.\\ & \quad\left. h(z)+\sqrt{\epsilon}< t< h(z)+2\sqrt{\epsilon}, ~ b_2(z,t)<\tau< b_2(z,t)+q{\rm e}^{-\alpha/\sqrt{\epsilon}}\right\}. \end{align*}

Note that $\gamma _\epsilon$ is uniformly bounded in $\epsilon$. Therefore we have

(3.7)\begin{equation} J_\epsilon(\rho_\epsilon,\Omega_{\Sigma_{h+2\sqrt{\epsilon}}}{\setminus}\Omega_{\Sigma_{h+\sqrt{\epsilon}}})=\mbox{o}(\epsilon). \end{equation}

Similarly we have

(3.8)\begin{equation} J_\epsilon(\rho_\epsilon,\Omega_{\Sigma_{h-\sqrt{\epsilon}}}{\setminus}\Omega_{\Sigma_{h-2\sqrt{\epsilon}}})=\mbox{o}(\epsilon). \end{equation}

Finally, we consider the integral $J_\epsilon (\rho _\epsilon,\,\Omega _{\Sigma _{h+\sqrt {\epsilon }}}{\setminus} \Omega _{\Sigma _{h-\sqrt {\epsilon }}})$. We have

\begin{align*} & J_\epsilon(\rho_\epsilon,\Omega_{\Sigma_{h+\sqrt{\epsilon}}}{\setminus}\Omega_{\Sigma_{h-\sqrt{\epsilon}}})\\ & \quad=\int_{\Omega_{\Sigma_{h+\sqrt{\epsilon}}}{\setminus}\Omega_{\Sigma_{h-\sqrt{\epsilon}}}}\frac{\epsilon}{2}a\left|\nabla_g W\left(z,t,\frac{t-h(z)}{\epsilon}\right)\right|^{2}\\ & \qquad+\frac{1}{\epsilon}F\left(z,t,W\left(z,t,\frac{t-h(z)}{\epsilon}\right)\right){\rm d}V_{\bar{g}}{\rm d}t \\ & \quad=\int_{\Omega_{\Sigma_{h+\sqrt{\epsilon}}}{\setminus}\Omega_{\Sigma_{h-\sqrt{\epsilon}}}}\frac{\epsilon}{2}a\left\{\left|\nabla_{\bar{g}} W\left(z,t,\frac{t-h(z)}{\epsilon}\right)\right|^{2}(1+\mbox{O}(t)) \right.\\ & \qquad\left. +\left[\partial_2W+\frac{1}{\epsilon}\partial_3W\right]^{2}\right\} +\frac{1}{\epsilon}F\left(z,t,W\left(z,t,\frac{t-h(z)}{\epsilon}\right)\right){\rm d}V_{\bar{g}}{\rm d}t, \end{align*}

where we used the formula $|\nabla _g v(z,\,t)|^{2}=|\nabla _{\bar {g}} v(z,\,t)|^{2}(1+\mbox {O}(t))+(\partial _tv(z,\,t))^{2}$. Then

\begin{align*} & J_\epsilon(\rho_\epsilon,\Omega_{\Sigma_{h+\sqrt{\epsilon}}}{\setminus}\Omega_{\Sigma_{h-\sqrt{\epsilon}}})\\ & \quad=\int_{\Omega_{\Sigma_{h+\sqrt{\epsilon}}}{\setminus}\Omega_{\Sigma_{h-\sqrt{\epsilon}}}}\frac{\epsilon}{2}a(z,t)\left\{\left|\nabla_{\bar{g}} W\left(z,t,\frac{t-h(z)}{\epsilon}\right)\right|^{2}(1+\mbox{O}(t))\right.\\ & \qquad\left. +\left[\partial_2W+\frac{1}{\epsilon}\sqrt{\frac{2F\left(z,t,W(z,t,\frac{t-h(z)}{\epsilon})\right)}{a(z,t)}} \right]^{2}\right\} \\ & \qquad+\frac{1}{\epsilon}F\left(z,t,W\Big(z,t,\frac{t-h(z)}{\epsilon}\Big)\right){\rm d}V_{\bar{g}}{\rm d}t\\ & \quad=\int_{\Omega_{\Sigma_{h+\sqrt{\epsilon}}}{\setminus}\Omega_{\Sigma_{h-\sqrt{\epsilon}}}}\frac{1}{2}\epsilon a\left[\left|\nabla_{\bar{g}} W\left(z,t,\frac{t-h(z)}{\epsilon}\right)\right|^{2}(1+\mbox{O}(t))+(\partial_2W)^{2}\right]\\ & \qquad+\partial_2W\sqrt{2aF\left(z,t,W(z,t,\frac{t-h(z)}{\epsilon})\right)} \\ & \qquad+\frac{2}{\epsilon}F\left(z,t,W\left(z,t,\frac{t-h(z)}{\epsilon}\right)\right){\rm d}V_{\bar{g}}{\rm d}t. \end{align*}

Note that

\[ \displaystyle\int_{\Omega_{\Sigma_{h+\sqrt{\epsilon}}}{\setminus}\Omega_{\Sigma_{h-\sqrt{\epsilon}}}}\displaystyle\frac{1}{2}\epsilon a\left[\left|\nabla_{\bar{g}} W\left(z,t,\frac{t-h(z)}{\epsilon}\right)\right|^{2}(1+\mbox{O}(t))+(\partial_2W)^{2}\right]=\mbox{o}(1), \]

in virtue of the properties of the solution $W$ of (2.2) and the fact that $\|\nabla _{\bar {g}} h\|_{L^{\infty }(\Sigma _0)}=\mbox {o}(\epsilon ^{{1}/{4}})$. The term $\partial _2W\sqrt {2aF}$ is bounded in $\Omega _{\Sigma _{h+\sqrt {\epsilon }}}{\setminus} \Omega _{\Sigma _{h-\sqrt {\epsilon }}}$. Now, letting $\mu ={(t-h(z))}/{\epsilon }$ and so $t=t(z,\,\mu )=h(z)+\epsilon \mu$, we have

(3.9)\begin{align} & J_\epsilon(\rho_\epsilon,\Omega_{\Sigma_{h+\sqrt{\epsilon}}}{\setminus}\Omega_{\Sigma_{h-\sqrt{\epsilon}}}) \nonumber\\ & \quad=\int_{\Omega_{\Sigma_{h+\sqrt{\epsilon}}}{\setminus}\Omega_{\Sigma_{h-\sqrt{\epsilon}}}}\frac{2}{\epsilon} F\left(z,t,W\left(z,t,\frac{t-h(z)}{\epsilon}\right)\right){\rm d}V_{\bar{g}}{\rm d}t +\mbox{o}(1)\nonumber\\ & \quad=\int_{-{1}/{\sqrt{\epsilon}}}^{{1}/{\sqrt{\epsilon}}}\int_{\Sigma_{0}}2F \left(z,h(z)+\epsilon\mu,W(z,h(z)+\epsilon\mu,\mu)\right)\nonumber\\ & \qquad\times(1+(h(z)+\epsilon\mu)\kappa+\mbox{o}(h(z)+\epsilon\mu)){\rm d}V_{\bar{g}}{\rm d}\mu +\mbox{o}(1). \end{align}

One has

(3.10)\begin{align} & \frac{{\rm d}}{{\rm d}\mu}\int_{\Sigma_{0}}\int_{b_1(z,h+\epsilon\mu)}^{W(z,h+\epsilon\mu;\mu)}\sqrt{\frac{1}{2}a(z,h+\epsilon\mu)F \left(z,h(z)+\epsilon\mu,\tau\right)}\nonumber\\ & \qquad\times(1+(h(z)+\epsilon\mu)\kappa+\mbox{o}(h(z)+\epsilon\mu)){\rm d} \tau {\rm d}V_{\bar{g}}\nonumber\\ & \quad=\int_{\Sigma_{0}}\int_{b_1(z,h+\epsilon\mu)}^{W(z,h+\epsilon\mu;\mu)}\frac{{\rm d}}{{\rm d}\mu}\left[\sqrt{\frac{1}{2}a(z,h+\epsilon\mu)F \left(z,h(z)+\epsilon\mu,\tau\right)} \right.\nonumber\\ & \qquad\left. \times(1+(h(z)+\epsilon\mu)\kappa+\mbox{o}(h(z)+\epsilon\mu))\right]{\rm d} \tau {\rm d}V_{\bar{g}} \nonumber\\ & \qquad+\int_{\Sigma_{0}}\sqrt{\frac{1}{2}a(z,h+\epsilon\mu)F \left(z,h(z)+\epsilon\mu,W(z,h+\epsilon\mu;\mu)\right)} \nonumber\\ & \qquad\times(\epsilon\partial_2W+\partial_3W) (1+(h(z)+\epsilon\mu)\kappa+\mbox{o}(h(z)+\epsilon\mu)){\rm d}V_{\bar{g}} \end{align}

Note that

(3.11)\begin{align} & \sqrt{\frac{1}{2}a(z,h+\epsilon\mu)F\left(z,h(z)+\epsilon\mu,W(z,h+\epsilon\mu;\mu)\right)}\partial_3W(z,h+\epsilon\mu;\mu) \nonumber\\ & \quad =F\left(z,h(z)+\epsilon\mu,W(z,h+\epsilon\mu;\mu)\right). \end{align}

By (3.10) and (3.11) we have

(3.12)\begin{align} & \int_{-{1}/{\sqrt{\epsilon}}}^{{1}/{\sqrt{\epsilon}}}\int_{\Sigma_{0}}2F \left(z,h(z)+\epsilon\mu,W(z,h(z)+\epsilon\mu,\mu)\right) \nonumber\\ & \qquad\times(1+(h(z)+\epsilon\mu)\kappa+\mbox{o}(h(z)+\epsilon\mu)){\rm d}V_{\bar{g}}{\rm d}\mu \nonumber\\ & \quad=\int_{\Sigma_{0}}\int_{b_1(z,h+\sqrt{\epsilon})}^{W(z,h+\sqrt{\epsilon};{1}/{\sqrt{\epsilon}})}\sqrt{2a(z,h+\sqrt{\epsilon})F \left(z,h(z)+\sqrt{\epsilon},\tau\right)} \nonumber\\ & \qquad\times(1+(h(z)+\sqrt{\epsilon})\kappa+\mbox{o}(h(z)+\sqrt{\epsilon})){\rm d} \tau {\rm d}V_{\bar{g}}\nonumber\\ & \qquad-\int_{\Sigma_{0}}\int_{b_1(z,h-\sqrt{\epsilon})}^{W(z,h-\sqrt{\epsilon};-{1}/{\sqrt{\epsilon}})}\sqrt{2a(z,h-\sqrt{\epsilon})F \left(z,h(z)-\sqrt{\epsilon},\tau\right)}\nonumber\\ & \qquad\times(1+(h(z)-\sqrt{\epsilon})\kappa+\mbox{o}(h(z)-\sqrt{\epsilon})){\rm d} \tau {\rm d}V_{\bar{g}}\nonumber\\ & \qquad-2\int_{-{1}/{\sqrt{\epsilon}}}^{{1}/{\sqrt{\epsilon}}}I_{1,\epsilon}{\rm d}\mu -2\int_{-{1}/{\sqrt{\epsilon}}}^{{1}/{\sqrt{\epsilon}}}I_{2,\epsilon}{\rm d}\mu, \end{align}

where

\begin{align*} I_{1,\epsilon}& =\int_{\Sigma_{0}}\int_{b_1(z,h+\epsilon\mu)}^{W(z,h+\epsilon\mu;\mu)}\frac{{\rm d}}{{\rm d}\mu}\left[\sqrt{\frac{1}{2}a(z,h+\epsilon\mu)F \left(z,h(z)+\epsilon\mu,\tau\right)}\right.\\ & \quad\left. \times(1+(h(z)+\epsilon\mu)\kappa+\mbox{o}(h(z)+\epsilon\mu))\right]{\rm d} \tau {\rm d}V_{\bar{g}}, \\ I_{2,\epsilon}& =\epsilon\int_{\Sigma_{0}}\sqrt{\frac{1}{2}a(z,h+\epsilon\mu)F \left(z,h(z)+\epsilon\mu,W(z,h+\epsilon\mu;\mu)\right)}\partial_2W \\ & \quad\times (1+(h(z)+\epsilon\mu)\kappa+\mbox{o}(h(z)+\epsilon\mu)){\rm d}V_{\bar{g}}. \end{align*}

Plainly

(3.13)\begin{equation} \int_{-{1}/{\sqrt{\epsilon}}}^{{1}/{\sqrt{\epsilon}}}I_{2,\epsilon}{\rm d}\mu=\mbox{O}(\sqrt{\epsilon}). \end{equation}

Recalling $t=t(z,\,\mu )=h(z)+\epsilon \mu$, we have

\begin{align*} & \frac{{\rm d}}{{\rm d}\mu}\left[\sqrt{\frac{1}{2}a(z,h+\epsilon\mu)F \left(z,h(z)+\epsilon\mu,\tau\right)} \right.\\ & \qquad\left.\times(1+(h(z)+\epsilon\mu)\kappa+\mbox{o}(h(z)+\epsilon\mu))\right] \\ & \quad=\varepsilon\frac{{\rm d}}{{\rm d}t}\left[\sqrt{\frac{1}{2}a(z,t)F \left(z,t,\tau\right)}(1+t\kappa+\mbox{o}(t))\right]. \end{align*}

Hence

(3.14)\begin{equation} \int_{-{1}/{\sqrt{\epsilon}}}^{{1}/{\sqrt{\epsilon}}}I_{1,\epsilon}{\rm d}\mu=\mbox{O}(\sqrt{\epsilon}). \end{equation}

From (3.9), (3.12), (3.13) and (3.14) we obtain

(3.15)\begin{align} & \lim_{\epsilon\rightarrow 0}J_\epsilon(\rho_\epsilon,\Omega_{\Sigma_{h+\sqrt{\epsilon}}}{\setminus}\Omega_{\Sigma_{h-\sqrt{\epsilon}}})\nonumber\\ & \quad=\int_{\Sigma_{0}}\int_{b_1(z,h(z))}^{b_2(z,h(z))}\sqrt{2a(z,h)F \left(z,h(z),\tau\right)}(1+h(z)\kappa+\mbox{o}(h(z))){\rm d} \tau {\rm d}V_{\bar{g}}\nonumber\\ & \quad=\sqrt{2}\int_{\Sigma_{0}}\Lambda(z,h(z))(1+h(z)\kappa+\mbox{o}(h(z))) {\rm d}V_{\bar{g}}\nonumber\\ & \quad=\sqrt{2}\int_{\Sigma_{h}}\Lambda {\rm d}V_{\bar{g}}. \end{align}

Combining the above claim and the assumption that $u_\epsilon$ is a family of $L^{1}$-local minimizer of $J_\epsilon$, we obtain

(3.16)\begin{equation} J_\epsilon(u_\epsilon)\leq J_\epsilon(\rho_\epsilon). \end{equation}

The upper bound estimate (3.4) follows from (3.16), (3.5), (3.6), (3.7), (3.8) and (3.15), where the relation $\Omega _+{\setminus} \Omega _{\Sigma _0}=\Omega _+$ is used again, since $\mathcal {Q}$ is simply connected.

For the case that $\mathcal {Q}$ is multiply connected, (3.5) becomes

\[ J_\epsilon(\rho_\epsilon,\Omega{\setminus}\Omega_{\Sigma_{h+2\sqrt{\epsilon}}}) =\displaystyle\frac{1}{\epsilon}\displaystyle\int_{\Omega_+{{\setminus}} \Omega_{\Sigma_0}}F(x,b_2(x)){\rm d}x+\mbox{O}(\epsilon), \]

and the same argument as that of the simply connected domain case gives the upper bound estimate (3.4).

Proof of theorem 1.3. Lemmas 3.1 and 3.2 give the desired results of theorem 1.3 after a simple proof by contradiction.

4. Global minimum

Given another smooth closed hypersurface $\tilde {\Sigma }\subset \mathcal {Q}$, similarly as the geometric ground in § 1, for some $\tilde {\delta }>0$, we define

\[ \tilde{S}=\{x\in\Omega: \mbox{dist}(x,\tilde{\Sigma})<2\tilde{\delta}\}, \tilde{\Upsilon}=[{-}2\tilde{\delta},2\tilde{\delta}]. \]

We parameterize elements $x\in \tilde {S}$ using their closest point $z$ in $\tilde {\Sigma }$ and their distance $t$. Define the diffeomorphism $\tilde {\Gamma }: \tilde {\Sigma }\times \tilde {\Upsilon }\rightarrow \tilde {S}$ by

\[ \tilde{\Gamma}(z,t)=z+t\mathbf{\tilde{n}}(z). \]

Letting $\tilde {\varphi }$ be the corresponding immersion into $\mathbb {R}^{N}$, we have

\[ \begin{cases} \displaystyle\frac{\partial \tilde{\Gamma}}{\partial z_i}(z,t)=\frac{\partial \tilde{\varphi}}{\partial z_i}(z)+t\tilde{\kappa}^{j}_i(z)\frac{\partial \tilde{\varphi}}{\partial z_j}(z) & \mbox{for}~ i=1, \ldots, N-1,\\ \displaystyle\frac{\partial \tilde{\Gamma}}{\partial t}(z,t)=\mathbf{\tilde{n}}(z).\\ \end{cases} \]

Let also $(\bar {\tilde {g}}_{ij})_{ij}$ be the coefficients of the metric on $\tilde {\Sigma }$ in the above coordinates $z$. Then, letting $\tilde {g}$ denote the metric on $\Omega$ induced by $\mathbb {R}^{N}$, we have

\[ \tilde{g}_{IJ}= \left( \begin{array}{cc} \{\tilde{g}_{ij}\} & 0\\ 0 & 1 \end{array} \right), \]

where

\[ \tilde{g}_{ij}=\bar{\tilde{g}}_{ij}+t(\tilde{\kappa}^{m}_i\bar{\tilde{g}}_{mj}+ \tilde{\kappa}^{n}_j\bar{\tilde{g}}_{in})+t^{2}\tilde{\kappa}^{m}_i\tilde{\kappa}^{n}_j\bar{\tilde{g}}_{mn}. \]

We have also

\[ \det \tilde{g}=\det \bar{\tilde{g}} [1+2t\tilde{\kappa}^{i}_i]+\mbox{o}(t)=:\det \bar{\tilde{g}} [1+2t\tilde{\kappa}]+\mbox{o}(t), \]

and

\[ {\rm d}V_{\tilde{g}}=\sqrt{\det \tilde{g}}{\rm d}z{\rm d}t=(1+t\tilde{\kappa}+\mbox{o}(t))\sqrt{\det \bar{\tilde{g}}}{\rm d}z{\rm d}t=(1+t\tilde{\kappa}+\mbox{o}(t)){\rm d}V_{\bar{\tilde{g}}}{\rm d}t. \]

For $h$ satisfying $\|h\|_{L^{\infty }(\tilde {\Sigma })}\leq 2\tilde {\delta }$, we define the perturbed closed hypersurface

\[ \tilde{\Sigma}_h:=\{\tilde{\Gamma}(z,h(z)): z\in\tilde{\Sigma}\}. \]

Lemma 4.1 Assume that $u_\epsilon$ is a family of global minimizer of $\bar {J}_\epsilon,$ we have

(4.1)\begin{equation} \bar{J}_\epsilon( u_\epsilon)\leq \sqrt{2}\int_{\tilde{\Sigma}}\Lambda {\rm d}V_{\bar{\tilde{g}}}+\int_{\Omega_+{{\setminus}} \Omega_{\tilde{\Sigma}}}\frac{1}{\epsilon}F(x,b_2(x)){\rm d}x+\mbox{o}(1). \end{equation}

Proof. First, we also assume that $\mathcal {Q}$ is simply connected. Similar to that of § 3 we define $\tilde {\rho }_\epsilon : \Omega \rightarrow \mathbb {R}$ by

\[ \tilde{\rho}_\epsilon(x)= \begin{cases} b_2(x), & x\in\Omega{\setminus}\Omega_{\tilde{\Sigma}_{\tilde{\delta}}},\\ b_\epsilon(z,t;t), & x=\tilde{\varphi}(z)+t\mathbf{\tilde{n}}(z)\in \Omega_{\tilde{\Sigma}_{\tilde{\delta}}}{\setminus}\Omega_{\tilde{\Sigma}_{-\tilde{\delta}}},\\ b_1(x), & x\in\Omega_{\tilde{\Sigma}_{-\tilde{\delta}}}.\\ \end{cases} \]

Decompose

\begin{align*} \bar{J}_\epsilon(\tilde{\rho}_\epsilon)& =\bar{J}_\epsilon(\tilde{\rho}_\epsilon,\Omega{\setminus}\Omega_{\tilde{\Sigma}_{2\sqrt{\epsilon}}}) +\bar{J}_\epsilon(\tilde{\rho}_\epsilon,\Omega_{\tilde{\Sigma}_{2\sqrt{\epsilon}}}{\setminus}\Omega_{\tilde{\Sigma}_{\sqrt{\epsilon}}})\\ & \quad+\bar{J}_\epsilon(\tilde{\rho}_\epsilon,\Omega_{\tilde{\Sigma}_{\sqrt{\epsilon}}}{\setminus}\Omega_{\tilde{\Sigma}_{-\sqrt{\epsilon}}}) +\bar{J}_\epsilon(\tilde{\rho}_\epsilon,\Omega_{\tilde{\Sigma}_{-\sqrt{\epsilon}}}{\setminus}\Omega_{\tilde{\Sigma}_{{-}2\sqrt{\epsilon}}}) \\ & \quad+\bar{J}_\epsilon(\tilde{\rho}_\epsilon,\Omega_{\tilde{\Sigma}_{{-}2\sqrt{\epsilon}}}). \end{align*}

Similar to that of (3.5), we have

(4.2)\begin{equation} \bar{J}_\epsilon(\tilde{\rho}_\epsilon,\Omega{\setminus}\Omega_{\tilde{\Sigma}_{2\sqrt{\epsilon}}}) =\frac{1}{\epsilon}\int_{\Omega_+}F(x,b_2(x)){\rm d}x+\mbox{O}(\epsilon), \end{equation}

and

(4.3)\begin{equation} \bar{J}_\epsilon(\tilde{\rho}_\epsilon,\Omega_{\tilde{\Sigma}_{{-}2\sqrt{\epsilon}}})=\mbox{O}(\epsilon). \end{equation}

We have

\[ \bar{J}_\epsilon(\tilde{\rho}_\epsilon,\Omega_{\tilde{\Sigma}_{2\sqrt{\epsilon}}}{\setminus}\Omega_{\tilde{\Sigma}_{\sqrt{\epsilon}}})= \displaystyle\frac{1}{\epsilon}\displaystyle\int_{\Omega_{\tilde{\Sigma}_{2\sqrt{\epsilon}}}{\setminus}\Omega_{\tilde{\Sigma}_{\sqrt{\epsilon}}}} F(x,\tilde{\rho}_\epsilon(x)){\rm d}x+\mbox{o}(\epsilon). \]

Using that $F(x,\,b_2(x))=0=F_u(x,\,b_2(x))$ and $F_{uu}(x,\,b_2(x))>0$ again, we have

\begin{align*} & \frac{1}{\epsilon}\int_{\Omega_{\tilde{\Sigma}_{2\sqrt{\epsilon}}}{\setminus}\Omega_{\tilde{\Sigma}_{\sqrt{\epsilon}}}} F(x,\tilde{\rho}_\epsilon(x)){\rm d}x\\ & \quad=\frac{1}{\epsilon}\int_{\sqrt{\epsilon}}^{2\sqrt{\epsilon}}\int_{\tilde{\Sigma}_{t}} F\left(z,t,\left[b_2(z,t)-W\left(z,t;\frac{1}{\sqrt{\epsilon}}\right)\right]\frac{t-2\sqrt{\epsilon}}{\sqrt{\epsilon}}\right.\\ & \qquad\left. +b_2(z,t)\right){\rm d}V_{\bar{\tilde{g}}}{\rm d}t \\ & \quad\leq \frac{1}{\epsilon}\int_{\sqrt{\epsilon}}^{2\sqrt{\epsilon}}\int_{\tilde{\Sigma}_{t}} F\left(z,t,q_1{\rm e}^{-\alpha/\sqrt{\epsilon}}+b_2(z,t)\right){\rm d}V_{\bar{\tilde{g}}}{\rm d}t \\ & \quad\leq\frac{1}{\epsilon}\int_{\sqrt{\epsilon}}^{2\sqrt{\epsilon}}q_1{\rm e}^{-\alpha/\sqrt{\epsilon}}\tilde{\gamma}_\epsilon {\rm d}t \\ & \quad=\mbox{o}(\epsilon), \end{align*}

where

\begin{align*} \nonumber \tilde{\gamma}_\epsilon& :=\sup\left\{\int_{\tilde{\Sigma}_{\mu}}F_u(z,\mu,\tau){\rm d}V_{\bar{\tilde{g}}}: \sqrt{\epsilon}<\mu< 2\sqrt{\epsilon},\right.\\ & \quad\left. b_2(z,\mu)<\tau< b_2(z,\mu)+q_1{\rm e}^{-\alpha/\sqrt{\epsilon}}\right\}. \end{align*}

Therefore, we have

(4.4)\begin{equation} \bar{J}_\epsilon(\tilde{\rho}_\epsilon,\Omega_{\tilde{\Sigma}_{2\sqrt{\epsilon}}}{\setminus}\Omega_{\tilde{\Sigma}_{\sqrt{\epsilon}}}) =\mbox{o}(\epsilon). \end{equation}

Similarly we have

(4.5)\begin{equation} \bar{J}_\epsilon(\tilde{\rho}_\epsilon,\Omega_{\tilde{\Sigma}_{-\sqrt{\epsilon}}}{\setminus}\Omega_{\tilde{\Sigma}_{{-}2\sqrt{\epsilon}}})=\mbox{o}(\epsilon). \end{equation}

For the integral $\bar {J}_\epsilon (\tilde {\rho }_\epsilon,\,\Omega _{\tilde {\Sigma }_{\sqrt {\epsilon }}}{\setminus} \Omega _{\tilde {\Sigma }_{-\sqrt {\epsilon }}})$, we have

\begin{align*} & \bar{J}_\epsilon(\tilde{\rho}_\epsilon,\Omega_{\tilde{\Sigma}_{\sqrt{\epsilon}}}{\setminus}\Omega_{\tilde{\Sigma}_{-\sqrt{\epsilon}}})\\ & \quad=\int_{\Omega_{\tilde{\Sigma}_{\sqrt{\epsilon}}}{\setminus}\Omega_{\tilde{\Sigma}_{-\sqrt{\epsilon}}}}\frac{\epsilon}{2}a\left|\nabla_{\tilde{g}} W\left(z,t,\frac{t}{\epsilon}\right)\right|^{2}+\frac{1}{\epsilon}F\left(z,t,W\left(z,t,\frac{t} {\epsilon}\right)\right) \\ & \quad=\int_{\Omega_{\tilde{\Sigma}_{\sqrt{\epsilon}}}{\setminus}\Omega_{\tilde{\Sigma}_{-\sqrt{\epsilon}}}} \frac{\epsilon}{2}a\left\{\left|\nabla_{\bar{\tilde{g}}} W\left(z,t,\frac{t}{\epsilon}\right)\right|^{2}(1+\mbox{O}(t)) \right.\\ & \qquad\left. +\left[\partial_2W+\frac{1}{\epsilon}\partial_3W\right]^{2}\right\} +\frac{1}{\epsilon}F\left(z,t,W\left(z,t,\frac{t}{\epsilon}\right)\right){\rm d}V_{\bar{\tilde{g}}}{\rm d}t \\ & \quad=\int_{\Omega_{\tilde{\Sigma}_{\sqrt{\epsilon}}}{\setminus}\Omega_{\tilde{\Sigma}_{-\sqrt{\epsilon}}}}\frac{\epsilon}{2}a \left\{\left|\nabla_{\bar{\tilde{g}}} W\left(z,t,\frac{t}{\epsilon}\right)\right|^{2}(1+\mbox{O}(t))\right.\\ & \qquad\left. +\left[\partial_2W+\frac{1}{\epsilon}\sqrt{\frac{2F}{a}} \right]^{2}\right\} +\frac{1}{\epsilon}F\left(z,t,W\left(z,t,\frac{t}{\epsilon}\right)\right)\\ & \quad=\int_{\Omega_{\tilde{\Sigma}_{\sqrt{\epsilon}}}{\setminus}\Omega_{\tilde{\Sigma}_{-\sqrt{\epsilon}}}}\frac{1}{2}\left\{\epsilon a\left[\left|\nabla_{\bar{\tilde{g}}} W\left(z,t,\frac{t}{\epsilon}\right)\right|^{2}(1+\mbox{O}(t))+(\partial_2W)^{2}\right] \right.\\ \end{align*}

Note that

\[ \epsilon a\left[\left|\nabla_{\bar{\tilde{g}}} W\left(\tilde{z},\tilde{t},\displaystyle\frac{\tilde{t}}{\epsilon}\right)\right|^{2}(1+\mbox{O}(t))+(\partial_2W)^{2}\right] \]

is bounded in $\Omega _{\Sigma _{h+\sqrt {\epsilon }}}{\setminus} \Omega _{\Sigma _{h-\sqrt {\epsilon }}}$ in virtue of the properties of the solution $W$ of (2.2). Hence, letting $\mu =\frac {t}{\epsilon }$ and so $t=t(\mu )=\epsilon \mu$, we have

(4.6)\begin{align} & \bar{J}_\epsilon(\tilde{\rho}_\epsilon,\Omega_{\tilde{\Sigma}_{\sqrt{\epsilon}}}{\setminus}\Omega_{\tilde{\Sigma}_{-\sqrt{\epsilon}}})\nonumber\\ & \quad=\int_{-{1}/{\sqrt{\epsilon}}}^{{1}/{\sqrt{\epsilon}}}\int_{\tilde{\Sigma}}2F \left(z,\epsilon\mu,W(z,\epsilon\mu,\mu)\right)(1+\epsilon\mu\tilde{\kappa}+\mbox{o}(\epsilon\mu)){\rm d}V_{\bar{\tilde{g}}}{\rm d}\mu +\mbox{O}(\sqrt{\epsilon}). \end{align}

One has

(4.7)\begin{align} & \frac{{\rm d}}{{\rm d}\mu}\int_{\tilde{\Sigma}}\int_{b_1(z,\epsilon\mu)}^{W(z,\epsilon\mu;\mu)}\sqrt{\frac{1}{2}a(z,\epsilon\mu)F \left(z,\epsilon\mu,\tau\right)}(1+\epsilon\mu\tilde{\kappa}+\mbox{o}(\epsilon\mu)){\rm d} \tau {\rm d}V_{\bar{\tilde{g}}} \nonumber\\ & \quad=\int_{\tilde{\Sigma}}\int_{b_1(z,\epsilon\mu)}^{W(z,\epsilon\mu;\mu)}\frac{{\rm d}}{{\rm d}\mu}\left[\sqrt{\frac{1}{2}a(z,\epsilon\mu)F \left(z,\epsilon\mu,\tau\right)}(1+\epsilon\mu\tilde{\kappa}+\mbox{o}(\epsilon\mu))\right]{\rm d} \tau {\rm d}V_{\bar{\tilde{g}}} \nonumber\\ & \qquad+\int_{\tilde{\Sigma}}\sqrt{\frac{1}{2}a(z,\epsilon\mu)F \left(z,\epsilon\mu,W(z,\epsilon\mu;\mu))\right)} \nonumber\\ & \qquad\times(\epsilon\partial_2W+\partial_3W) (1+\epsilon\mu\tilde{\kappa}+\mbox{o}(\epsilon\mu)){\rm d}V_{\bar{\tilde{g}}} \end{align}

Note that

(4.8)\begin{align} & \sqrt{\frac{1}{2}a(z,\epsilon\mu)F\left(z,\epsilon\mu,W(z,\epsilon\mu;\mu)\right)}\partial_3W(z,h+\epsilon\mu;\mu) \nonumber\\ & \quad=F\left(z,\epsilon\mu,W(z,\epsilon\mu;\mu)\right). \end{align}

By (4.7) and (4.8) we have

(4.9)\begin{align} & \int_{-{1}/{\sqrt{\epsilon}}}^{{1}/{\sqrt{\epsilon}}}\int_{\tilde{\Sigma}}2F \left(z,\epsilon\mu,W(z,\epsilon\mu,\mu)\right)(1+\epsilon\mu\tilde{\kappa}+\mbox{o}(\epsilon\mu)){\rm d}V_{\bar{\tilde{g}}}{\rm d}\mu \nonumber\\ & \quad=\int_{\tilde{\Sigma}}\int_{b_1(z,\sqrt{\epsilon})}^{W(z,\sqrt{\epsilon};{1}/{\sqrt{\epsilon}})}\sqrt{2a(z,\sqrt{\epsilon}) F\left(z,\sqrt{\epsilon},\tau\right)}(1+\sqrt{\epsilon}\tilde{\kappa}+\mbox{o}(\sqrt{\epsilon})){\rm d} \tau {\rm d}V_{\bar{\tilde{g}}} \nonumber\\ & \qquad-\int_{\tilde{\Sigma}}\int_{b_1(z,-\sqrt{\epsilon})}^{W(z,-\sqrt{\epsilon};-{1}/{\sqrt{\epsilon}})} \sqrt{2a(z,-\sqrt{\epsilon})F \left(z,-\sqrt{\epsilon},\tau\right)}(1-\sqrt{\epsilon}\tilde{\kappa}+\mbox{o}(\sqrt{\epsilon})){\rm d} \tau {\rm d}V_{\bar{\tilde{g}}}\nonumber\\ & \qquad-2\int_{-{1}/{\sqrt{\epsilon}}}^{{1}/{\sqrt{\epsilon}}}\tilde{I}_{1,\epsilon}{\rm d}\mu -2\int_{-{1}/{\sqrt{\epsilon}}}^{{1}/{\sqrt{\epsilon}}}\tilde{I}_{2,\epsilon}{\rm d}\mu, \end{align}

where

\begin{align*} \tilde{I}_{1,\epsilon}& =\int_{\tilde{\Sigma}}\int_{b_1(z,\epsilon\mu)}^{W(z,\epsilon\mu;\mu)}\frac{{\rm d}}{{\rm d}\mu} \left[\sqrt{\frac{1}{2}a(z,\epsilon\mu)F \left(z,\epsilon\mu,\tau\right)}(1+\epsilon\mu\tilde{\kappa}+\mbox{o}(\epsilon\mu))\right]{\rm d} \tau {\rm d}V_{\bar{\tilde{g}}}, \\ \tilde{I}_{2,\epsilon}& =\epsilon\int_{\tilde{\Sigma}}\sqrt{\frac{1}{2}a(z,\epsilon\mu)F \left(z,\epsilon\mu,W(z,\epsilon\mu;\mu))\right)}\partial_2W (1+\epsilon\mu\tilde{\kappa}+\mbox{o}(\epsilon\mu)){\rm d}V_{\bar{\tilde{g}}}. \end{align*}

Plainly

(4.10)\begin{equation} \int_{-{1}/{\sqrt{\epsilon}}}^{{1}/{\sqrt{\epsilon}}}\tilde{I}_{2,\epsilon}{\rm d}\mu=\mbox{O}(\sqrt{\epsilon}). \end{equation}

Recalling $t=t(\mu )=\epsilon \mu$, we have

\begin{align*} & \frac{{\rm d}}{{\rm d}\mu}\left[\sqrt{\frac{1}{2}a(z,\epsilon\mu)F \left(z,\epsilon\mu,\tau)\right)}(1+\epsilon\mu\tilde{\kappa}+\mbox{o}(\epsilon\mu))\right] \\ & \quad=\varepsilon\frac{{\rm d}}{{\rm d}t}\left[\sqrt{\frac{1}{2}a(\tilde{z},\tilde{t})F \left(z,t,\tau)\right)}(1+t\tilde{\kappa}+\mbox{o}(t))\right], \end{align*}

which yields

(4.11)\begin{equation} \int_{-{1}/{\sqrt{\epsilon}}}^{{1}/{\sqrt{\epsilon}}}\tilde{I}_{1,\epsilon}{\rm d}\mu=\mbox{O}(\sqrt{\epsilon}). \end{equation}

From (4.6), (4.9), (4.10) and (4.11) we obtain

(4.12)\begin{align} & \lim_{\epsilon\rightarrow 0}\bar{J}_\epsilon(\tilde{\rho}_\epsilon,\Omega_{\tilde{\Sigma}_{\sqrt{\epsilon}}}{\setminus} \Omega_{\tilde{\Sigma}_{-\sqrt{\epsilon}}})\nonumber\\ & \quad=\int_{\tilde{\Sigma}}\int_{b_1(z,0)}^{b_2(z,0)}\sqrt{2a(z,0)F \left(z,0,\tau)\right)}{\rm d} \tau {\rm d}V_{\bar{\tilde{g}}}\nonumber\\ & \quad=\sqrt{2}\int_{\tilde{\Sigma}}\Lambda(z,0) {\rm d}V_{\bar{\tilde{g}}}. \end{align}

The upper bound estimate (4.1) follows from (4.2), (4.3), (4.4), (4.5), (4.12) and the assumption that $\bar {J}_\epsilon (u_\epsilon )\leq \bar {J}_\epsilon (\tilde {\rho }_\epsilon )$, where the relation $\Omega _+{\setminus} \Omega _{\tilde {\Sigma }}=\Omega _+$ is used, since $\mathcal {Q}$ is simply connected.

For the case that $\mathcal {Q}$ is multiply connected, (4.2) becomes

\[ \bar{J}_\epsilon(\tilde{\rho}_\epsilon,\Omega{\setminus}\Omega_{\tilde{\Sigma}_{2\sqrt{\epsilon}}}) =\displaystyle\frac{1}{\epsilon}\displaystyle\int_{\Omega_+{{\setminus}} \Omega_{\tilde{\Sigma}}}F(x,b_2(x)){\rm d}x+\mbox{O}(\epsilon), \]

and the same argument as that of the simply connected domain case gives the desired result.

On the other hand, from lemma (3.1) we have

(4.13)\begin{equation} \bar{J}_\epsilon( u_\epsilon)\geq \sqrt{2}\int_{\Sigma_0}\Lambda {\rm d}V_{\bar{g}}+\int_{\Omega_+{{\setminus}} \Omega_{\Sigma_0}}\frac{1}{\epsilon}F(x,b_2(x)){\rm d}x+\mbox{o}(1). \end{equation}

Proof of theorem 1.4. Recall the assumption that $\Omega _+\backslash \Omega _{\Sigma }=\Omega _+\backslash \Omega _{\Sigma _0}$ for any closed smooth $(N-1)$-dimensional nontrivial surface $\Sigma \subset \mathcal {Q}$. Combining this, lemma 4.1 and (4.13) we obtain the desired results of theorem 1.4.

To find the locations of the interfaces of interior layers to $L^{1}$-local and global maximizers of the associated energy functional, or even to general layer solutions, seems to be an interesting question. What about $H^{1}$-local and global minimizers or maximizers is also deserved to be studied.