#### Vol. 13, No. 1, 2020

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Capillary surfaces arising in singular perturbation problems

### Aram L. Karakhanyan

Vol. 13 (2020), No. 1, 171–200
##### 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

in the unit ball ${B}_{1}$ as $\epsilon \to 0$. Here ${\beta }_{\epsilon }\left(t\right)=\left(1∕\epsilon \right)\beta \left(t∕\epsilon \right)\ge 0$, $\beta \in {C}_{0}^{\infty }\left[0,1\right]$, ${\int }_{0}^{1}\beta \left(t\right)\phantom{\rule{0.3em}{0ex}}dt=M>0$, 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 $g$.

##### Keywords
singular perturbation problem, free boundary regularity, capillary surfaces, global solutions
##### Mathematical Subject Classification 2010
Primary: 49Q05, 35R35, 35B25
##### Milestones
Revised: 5 September 2018
Accepted: 19 December 2018
Published: 6 January 2020
##### Authors
 Aram L. Karakhanyan School of Mathematics The University of Edinburgh Edinburgh United Kingdom