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2020 Capillary surfaces arising in singular perturbation problems
Aram L. Karakhanyan
Anal. PDE 13(1): 171-200 (2020). 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 2 and 3 arising as limits of the singular perturbation problem

u ε ( x ) = β ε ( u ε )  in  B 1 , | u ε | 1  in  B 1 ,

in the unit ball B1 as ε0. Here βε(t)=(1ε)β(tε)0, βC0[0,1], 01β(t)dt=M>0, is an approximation of the Dirac measure and ε>0. The limit functions u= limεj0uεj of uniformly converging sequences {uεj} 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 {u0>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 uε based on a computation of J. Spruck. It implies that any blow-up u0 of u either vanishes identically or is a homogeneous function of degree 1, that is, u0=rg(σ), σSN1, 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 u0 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 2M. In particular, we show that u0:S23 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.

Citation

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Aram L. Karakhanyan. "Capillary surfaces arising in singular perturbation problems." Anal. PDE 13 (1) 171 - 200, 2020. https://doi.org/10.2140/apde.2020.13.171

Information

Received: 14 February 2018; Revised: 5 September 2018; Accepted: 19 December 2018; Published: 2020
First available in Project Euclid: 16 January 2020

zbMATH: 07171991
MathSciNet: MR4047644
Digital Object Identifier: 10.2140/apde.2020.13.171

Subjects:
Primary: 35B25 , 35R35 , 49Q05

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

Rights: Copyright © 2020 Mathematical Sciences Publishers

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