Capillary surfaces arising in singular perturbation problems

Karakhanyan, Aram L.

doi:10.2140/apde.2020.13.171

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Capillary surfaces arising in singular perturbation problems

Aram L. Karakhanyan

Vol. 13 (2020), No. 1, 171–200

DOI: 10.2140/apde.2020.13.171

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

\{\begin{matrix} △ u_{ε} (x) = β_{ε} (u_{ε}) & in B_{1}, \\ | u_{ε} | \leq 1 & in B_{1}, \end{matrix}

in the unit ball $B_{1}$ as $ε \to 0$ . Here $β_{ε} (t) = (1 ∕ ε) β (t ∕ ε) \geq 0$ , $β \in C_{0}^{\infty} [0, 1]$ , $\int_{0}^{1} β (t) d t = M > 0$ , is an approximation of the Dirac measure and $ε > 0$ . The limit functions $u = lim_{ε_{j} \to 0} u_{ε_{j}}$ of uniformly converging sequences ${u_{ε_{j}}}$ 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 $\partial {u_{0} > 0}$ 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 $u_{ε}$ 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, $u_{0} = r g (σ)$ , $σ \in S^{N - 1}$ , in spherical coordinates $(r, θ)$ . 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 $\sqrt{2 M}$ . In particular, we show that $\nabla u_{0} : S^{2} \to ℝ^{3}$ 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 $g$ .

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Keywords

singular perturbation problem, free boundary regularity, capillary surfaces, global solutions

Mathematical Subject Classification 2010

Primary: 49Q05, 35R35, 35B25

Milestones

Received: 14 February 2018

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

Vol. 13, No. 1, 2020