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

TL;DR

This paper establishes the uniqueness and regularity of free-boundaries at isolated singularities for the Alt-Caffarelli functional, introducing new inequalities and connecting epiperimetric and Łojasiewicz inequalities.

Contribution

It proves a (log-)epiperimetric inequality for traces near cones with isolated singularities, leading to regularity results at singular points in the one-phase Bernoulli problem.

Findings

Proves uniqueness of blow-ups at isolated singularities.

Establishes $C^{1, ext{log}}$-regularity of the free boundary.

Connects epiperimetric inequalities with Łojasiewicz inequality.

Abstract

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 \L ojasiewicz inequality, and, to our knowledge, provides the first regularity result at singular points in the one-phase Bernoulli problem.