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

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.

singular perturbation problem, free boundary regularity, capillary surfaces, global solutions
Mathematical Subject Classification 2010
Primary: 49Q05, 35R35, 35B25
Received: 14 February 2018
Revised: 5 September 2018
Accepted: 19 December 2018
Published: 6 January 2020
Aram L. Karakhanyan
School of Mathematics
The University of Edinburgh
United Kingdom