1. Introduction
We study the following inhomogeneous transition problems
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
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
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
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
We introduce the set
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
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.
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
(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
\[ \hat{J}_\epsilon(u):=\begin{cases} \bar{J}_\epsilon(u), & u\in H^{1}(0,1),\\ \infty, & u\in L^{1}(0,1){\setminus} H^{1}(0,1).\\ \end{cases} \](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
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
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
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
where
We have, formally
where
There holds
and hence
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
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
We also introduce
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
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
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
2. Preliminaries
We first recall definition of functions with bounded variation and a property to be used. The interested reader is referred to [Reference Giusti19].
Definition 2.1 A function $\phi \in L^{1}(\Omega )$ is said to have bounded variation in $\Omega$ if
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
Consider the following initial value problem
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
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 [Reference Sotomayor29] (see also [Reference Ahmad and Ambrosetti1, Reference Sternberg30]).
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
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
Set
then we have
If we denote
then
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
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
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
Proof. First we also assume that $\mathcal {Q}$ is simply connected. We borrow the idea of [Reference Sternberg30] (see also [Reference Sônego28]) to define a sequence of functions $b_\epsilon (z,\,t;\tau ): \Sigma _0\times \Upsilon \times \Upsilon \rightarrow \mathbb {R}$
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
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
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
For the first integral of the right hand side we have
Analogously,
We have
All in all we obtain that $\rho _\epsilon \rightarrow \rho _0$ in $L^{1}(\Omega )$ as $\epsilon \rightarrow 0$.
We decompose
From the definition of $\rho _\epsilon$, we have
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
We have
Recalling that $F(x,\,b_2(x))=0=F_u(x,\,b_2(x))$ and $F_{uu}(x,\,b_2(x))>0$, we have
where
Note that $\gamma _\epsilon$ is uniformly bounded in $\epsilon$. Therefore we have
Similarly we have
Finally, we consider the integral $J_\epsilon (\rho _\epsilon,\,\Omega _{\Sigma _{h+\sqrt {\epsilon }}}{\setminus} \Omega _{\Sigma _{h-\sqrt {\epsilon }}})$. We have
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
Note that
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
One has
Note that
where
Plainly
Recalling $t=t(z,\,\mu )=h(z)+\epsilon \mu$, we have
Hence
From (3.9), (3.12), (3.13) and (3.14) we obtain
Combining the above claim and the assumption that $u_\epsilon$ is a family of $L^{1}$-local minimizer of $J_\epsilon$, we obtain
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
and the same argument as that of the simply connected domain case gives the upper bound estimate (3.4).
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
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
Letting $\tilde {\varphi }$ be the corresponding immersion into $\mathbb {R}^{N}$, we have
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
where
We have also
and
For $h$ satisfying $\|h\|_{L^{\infty }(\tilde {\Sigma })}\leq 2\tilde {\delta }$, we define the perturbed closed hypersurface
Lemma 4.1 Assume that $u_\epsilon$ is a family of global minimizer of $\bar {J}_\epsilon,$ we have
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
Decompose
Similar to that of (3.5), we have
and
We have
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
where
Therefore, we have
Similarly we have
For the integral $\bar {J}_\epsilon (\tilde {\rho }_\epsilon,\,\Omega _{\tilde {\Sigma }_{\sqrt {\epsilon }}}{\setminus} \Omega _{\tilde {\Sigma }_{-\sqrt {\epsilon }}})$, we have
Note that
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
One has
Note that
where
Plainly
Recalling $t=t(\mu )=\epsilon \mu$, we have
which yields
From (4.6), (4.9), (4.10) and (4.11) we obtain
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
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
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.