Capillary surfaces arising in singular perturbation problems

Abstract

We prove some Bernstein-type theorems for a class of stationary points of the Alt–Caffarelli functional in ${ℝ}^2$ and ${ℝ}^3$ arising as limits of the singular perturbation problem

$$\left\{\begin{array}{cc}△u_{\epsilon }\left(x\right)={\beta }_{\epsilon }\left(u_{\epsilon }\right)\hfill & \text{ in }{B}_1,\hfill \\ \left|u_{\epsilon }\right|\le 1\hfill & \text{ in }{B}_1,\hfill \\ \hfill \end{array}\right.$$

in the unit ball $B_1$ as $\epsilon \to 0$. Here ${\beta }_{\epsilon }\left(t\right)=\left(1∕\epsilon \right)\beta \left(t∕\epsilon \right)\ge 0$, $\beta \in C_0^{\infty }\left[0,1\right]$, ${\int }_0^1\beta \left(t\right)dt=M>0$, is an approximation of the Dirac measure and $\epsilon >0$. The limit functions $u=\underset{{\epsilon }_j\to 0}{lim}{u}_{{\epsilon }_j}$ of uniformly converging sequences $\left\{u_{{\epsilon }_j}\right\}$ solve a Bernoulli-type free boundary problem in some weak sense. Our approach has two novelties: First we develop a hybrid method for stratification of the free boundary $\partial \left\{u_0>0\right\}$ of blow-up solutions which combines some ideas and techniques of viscosity and variational theory. An important tool we use is a new monotonicity formula for the solutions $u_{\epsilon }$ based on a computation of J. Spruck. It implies that any blow-up $u_0$ of $u$ either vanishes identically or is a homogeneous function of degree 1, that is, $u_0=rg\left(\sigma \right)$, $\sigma \in {\mathbb{S}}^{N-1}$, in spherical coordinates $\left(r,\theta \right)$. In particular, this implies that in two dimensions the singular set is empty at the nondegenerate points, and in three dimensions the singular set of $u_0$ is at most a singleton. Second, we show that the spherical part $g$ is the support function (in Minkowski’s sense) of some capillary surface contained in the sphere of radius $\sqrt{2M}$. In particular, we show that $\nabla u_0:{\mathbb{S}}^2\to {ℝ}^3$ is an almost conformal and minimal immersion and the singular Alt–Caffarelli example corresponds to a piece of catenoid which is a unique ring-type stationary minimal surface determined by the support function $g$.