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 [Reference Allen and Cahn5].

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 [Reference Do Nascimento and Sônego12]). 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, [Reference Nakashima25] 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 [Reference Du and Gui13] generalized the results of [Reference Nakashima25] 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 [Reference Du and Lai15, Reference Du and Wang16, Reference Li and Nakashima21, Reference Yang and Yang32, Reference Zúñiga and Agudelo33]. For the corresponding fractional Laplacian, layer solutions are constructed in [Reference Du, Gui, Sire and Wei14]. For $a(x)\equiv 1$ and $f$ given by (1.3), there are many known existence results of transition layer solutions, see [Reference Alikakos and Bates2–Reference Alikakos and Simpson4, Reference Dancer and Yan6–Reference Do Nascimento and Sônego12, Reference Du and Wei17, Reference Fife and Greenlee18, Reference Mahmoudi, Malchiodi and Wei22, Reference Wei and Yang31].

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 [Reference Modica23] (see also [Reference Kohn and Sternberg20, Reference Modica and Mortola24]) 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 $\Lambda (x)$ [Reference Nakashima26], and, in higher-dimensional cases, the authors in [Reference Li and Nakashima21] 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 [Reference Nakashima and Tanaka27] 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 $(f_1)$–$(f_3)$, [Reference Sônego28] obtains the following results.

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

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