Calculus of Variations and Geometric Measure Theory

M. Engelstein - L. Spolaor - B. Velichkov

Uniqueness of the blow-up at isolated singularities for the Alt-Caffarelli functional

created by spolaor on 29 Jan 2018
modified by velichkov on 04 Nov 2019


Accepted Paper

Inserted: 29 jan 2018
Last Updated: 4 nov 2019

Journal: Duke Math. J.
Year: 2018


In this paper we prove uniqueness of blow-ups and $C^{1,\log}$-regularity for the free-boundary of minimizers of the Alt-Caffarelli functional at points where one blow-up has an isolated singularity. We do this by establishing a (log-)epiperimetric inequality for the Weiss energy for traces close to that of a cone with isolated singularity, whose free-boundary is graphical and smooth over that of the cone in the sphere. With additional assumptions on the cone, we can prove a classical epiperimetric inequality which can be applied to deduce a $C^{1,\alpha}$ regularity result. We also show that these additional assumptions are satisfied by the De Silva-Jerison-type cones, which are the only known examples of minimizing cones with isolated singularity. Our approach draws a connection between epiperimetric inequalities and the Lojasiewicz inequality, and, to our knowledge, provides the first regularity result at singular points of the Alt-Caffarelli functional.

Keywords: Regularity of free boundary