Limit Shapes of Local Minimizers for the Alt–Caffarelli Energy Functional in Inhomogeneous Media

Abstract

This paper considers the Alt–Caffarelli free boundary problem in a periodic medium. This is a convenient model for several interesting phenomena appearing in the study of contact lines on rough surfaces, pinning, hysteresis and the formation of facets. We show the existence of an interval of effective pinned slopes at each direction $e \in S^{d-1}$. In $d=2$ we characterize the regularity properties of the pinning interval in terms of the normal direction, including possible discontinuities at rational directions. These results require a careful study of the families of plane-like solutions available with a given slope. Using the same techniques we also obtain strong, in some cases optimal, bounds on the class of limit shapes of local minimizers in $d=2$, and preliminary results in $d \geqq 3$.

Appendix A. Augmented pinning problem as a limit of spatially homogeneous problems

In this section we derive the augmented pinning problem via a limit of regular spatially homogeneous pinning problems. This gives at least a plausibility argument for why $Q_{*,cont}$ and/or $Q^*_{cont}$ may be nontrivially different from the upper and lower semicontinuous envelopes of $Q_*$ and $Q^*$.

Consider a natural limiting procedure to derive (1.4), one might choose to regularize the jump discontinuities of $[Q_*,Q^*]$. It is natural to do this in a monotone way by an inf/sup convolution. We define the inf and sup convolving operators $\Box _{*,n}$ and $\Box ^*_{n}$ respectively on $LSC(S^{d-1})$ and $USC(S^{d-1})$

$$\begin{aligned} \Box _{*,n} f(e):= \inf _{e'\in S^{d-1}}\{ f(e') + n|e'-e|\} \ \hbox { and } \ \Box ^*_{n}f(e) = \sup _{e'\in S^{d-1}}\{ f(e') - n|e'-e|\}. \end{aligned}$$

Note that $\Box _{*,n} f$ and $\Box ^*_{n} f$ are Lipschitz continuous with constant n on $S^{d-1}$. The natural monotone approximation procedure would be to define

$$\begin{aligned} Q_{*,n}(e) = \Box _{*,n}Q_{*}(e) \ \hbox { and } \ Q^*_n(e) = \Box ^*_{n}Q^*_{n}(e). \end{aligned}$$

Basically we are regularizing the discontinuities of I(e), replacing by Lipschitz spikes. In this case it is straightforward to check that the minimal supersolution ${\underline{u}}_n$ and maximal subsolution ${\overline{u}}_n$ associated with $Q_{*,n}$ and $Q^*_n$ do converge, respectively, to the minimal supersolution ${\underline{u}}$ and maximal subsolution ${\overline{u}}$ of (10.2).

Now we consider a different approximation procedure which is not monotone. Assume that we are given $I(e) = [Q_*(e),Q^*(e)]$ and $I_{cont}(e) = [Q_{*,cont}(e),Q^*_{cont}(e)]$ satisfying the assumptions listed in Section 10.2. Define

$$\begin{aligned} Q_{*,m}(e)= & {} {\left\{ \begin{array}{ll} \Box ^*_{m}Q_{*,cont}(e) &{} e \hbox { irrational} \\ Q_{*}(e) &{} e \hbox { rational} \end{array}\right. } \ \hbox { and } \nonumber \\ Q^*_{m}(e)= & {} {\left\{ \begin{array}{ll} \Box _{*,m}Q^*_{cont}(e) &{} e \hbox { irrational} \\ Q^*(e) &{} e \hbox { rational} \end{array}\right. } \end{aligned}$$

(A.1)

and send $m \rightarrow \infty$. This isn’t really a regularization, $Q_{*,m}$ and $Q^*_{m}$ may still have jump discontinuities at rational directions, but one can think of regularizing again

$$\begin{aligned} Q_{*,m,n}(e)= \Box _{*,n}Q_{*,m}(e) \ \hbox { and } \ Q^*_{m,n}(e)= \Box ^*_{n}Q^*_{m}(e). \end{aligned}$$

and sending both $m,n \rightarrow \infty$ but with $m= o(n)$.

The pinning intervals $I_m(e)$ still converge as $m \rightarrow \infty$ pointwise to I(e), however the convergence is no longer monotone. Are all solutions of (1.4) achieved as a limit of solutions to (1.4)$_m$ for the approximating pinning intervals $[Q_{*,m}(e),Q^*_m(e)]$? It turns out that the answer is no, limits of solutions to (1.4)$_m$ actually satisfy a stronger condition in general.

Proposition A.1

Let $d=2$, U such that ${\mathbb {R}}^2{\setminus } U$ is convex. We refer to (10.2)$_m$ for problem (10.2) with pinning interval $[Q_{*,m},Q^*_m]$ as defined in (A.1).

• (i) Let $u_m$ be the minimal supersolution of (10.2)$_m$, then $u_m \rightarrow {\underline{u}}$ uniformly where ${\underline{u}}$ is the minimal supersolution of (10.4).

• (ii) Let $u_m$ be the maximal subsolution of (10.2)$_m$, then $u_m \rightarrow {\underline{u}}$ uniformly where ${\underline{u}}$ is the maximal subsolution of (10.4).

• (iii) Let u be a solution to (10.4) with convex support. Then there exists a sequence of solutions $u_m$ to (10.2)$_m$ such that $u_m \rightarrow u$ uniformly as $m \rightarrow \infty$.

Proof

We show convergence of the minimal supersolution and maximal subsolution. The last part follows as in Proposition 10.7.

First the minimal supersolutions. Suppose that $u_m \rightarrow u$ uniformly along some subsequence. By Theorem 10.4 we just need to check the supersolution and weak subsolution property for u. The supersolution property is easy because of the monotonicity $Q^*_m \nearrow Q^*$. The weak subsolution property is also easy because we only need to test with linear functions, the convergence $Q^*_m \rightarrow Q^*$ pointwise on $S^{d-1}$ is enough.

Now the maximal subsolutions, again we just need to check the subsolution and weakened supersolution condition. Suppose that $u_m \rightarrow u$ uniformly along some subsequence. The supersolution property in the limit is, for any $\varphi$ touching u from below at $x\in U \cap \partial \{u>0\}$ either $\Delta \varphi (x) \leqq 0$ or

$$\begin{aligned} |\nabla \varphi |(x) \leqq \limsup _{y \rightarrow x, m \rightarrow \infty } Q_{*,m}(\nabla \varphi (y)). \end{aligned}$$

Since $Q_{*,cont}$ is upper semicontinuous for any $\varepsilon >0$ there is a neighborhood N of $\nabla \varphi (x)$ such that for m sufficiently large and $e \in N$ we have $Q_{*,m}(e) \leqq Q_{*,cont}(\nabla \varphi ) +\varepsilon$. Thus

$$\begin{aligned} |\nabla \varphi |(x) \leqq Q_{*,cont}(\nabla \varphi (x)). \end{aligned}$$

Now we consider the weak subsolution condition, this is the interesting part. Suppose that $\varphi (x) = (p \cdot x)_+$ touches u from above at $0 \in \partial \{u>0\} \cap U$ in some domain $D \subset U$ with p rational. Then we can find $x_m \rightarrow 0$ such that $\varphi (x - x_m) = [p \cdot (x-x_m)]_+$ touches $u_m$ from above at $x_m \in \partial \{u_m>0\}$. Since $\{u_m>0\}$ is convex $|\nabla u_m|$ is defined and concave on the facet $\{p \cdot (x-x_m) = 0\} \cap \partial \{u_m>0\}$. For m sufficiently large the left and right limits of $Q_{*,m}$ at e are $Q_{*,cont}(e)$. We argue below that the facet must be a singleton $\{p \cdot (x-x_m) = 0\} \cap \partial \{u_m>0\}= \{x_m\}$. This means that given $r>0$ small enough that $B_r(x_m) \subset D$ and for $|q - p|$ sufficiently small $u_m> [q \cdot (x-x_m)]_+$ is compactly contained in $B_r(x_m)$ so for an appropriate choice of $x_m$ now $[q \cdot (x-x_m')]_+$ touches $u_m$ from above in $B_r(x_m)$ at $x_m' \in \partial \{u_m>0\}$. Therefore

$$\begin{aligned} |q| \geqq Q_{*,m}(q) \end{aligned}$$

and

$$\begin{aligned} |p| \geqq \lim _{m \rightarrow \infty }\lim _{q \rightarrow p, \ q \ne p}Q_{*,m}(q) = Q_{*,cont}(p). \end{aligned}$$

The case of irrational p is easy because of the correct monotonicity.

Now we argue that if $u_m$ solves (10.2) with convex support and the left and right limits of $Q_{*,m}$ at p agree, with value $Q_{*,cont}$ in this case, then the facet with normal p, call it $\Omega _p$, is trivial. This fact was used above. The argument is in Caffarelli–Lee [6, Lemma 3.5], but it is very brief so we repeat it here with more details. Suppose $\Omega _p$ is a non-trivial line segment, without loss $0 \in \Omega _p$. Then $|\nabla u_m|$ is concave on the facet and therefore must be identically equal to $Q_{*,cont}(p)$. Then extend u by reflection through $\Omega _p$ and subtract off the linear function $Q^*(p)x\cdot p$ to obtain a harmonic function v in an open domain ${\mathbb {R}}^2 \supset V \supset \Omega _p$ with $v = 0$ and $|\nabla v| = 0$ on $\Omega _p$. We identify ${\mathbb {R}}^2$ with the complex plane ${\mathbb {C}}$ and after rotation we can assume that $\Omega _p$ is a segment of the real line. Then

$$\begin{aligned} \varphi = v_y - iv_x \end{aligned}$$

is analytic in V and $\varphi = 0$ on $V \cap {\mathbb {R}}$. Thus $\varphi \equiv 0$ in V and u is linear with slope p in $\Omega _p$, this is not the case. $\square$