Vol. 13, No. 1, 2020

Download this article
Download this article For screen
For printing
Recent Issues

Volume 13
Issue 4, 945–1268
Issue 3, 627–944
Issue 2, 317–625
Issue 1, 1–316

Volume 12, 8 issues

Volume 11, 8 issues

Volume 10, 8 issues

Volume 9, 8 issues

Volume 8, 8 issues

Volume 7, 8 issues

Volume 6, 8 issues

Volume 5, 5 issues

Volume 4, 5 issues

Volume 3, 4 issues

Volume 2, 3 issues

Volume 1, 3 issues

The Journal
About the Journal
Editorial Board
Subscriptions
Editors’ Interests
Scientific Advantages
Submission Guidelines
Submission Form
Editorial Login
Ethics Statement
ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Author Index
To Appear
 
Other MSP Journals
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 2 and 3 arising as limits of the singular perturbation problem

uε(x) = βε(uε) in B1, |uε| 1  in B1,

in the unit ball B1 as ε 0. Here βε(t) = (1ε)β(tε) 0, β C0[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), σ SN1 , 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 : S2 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
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