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

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

arXiv comment37 pages. To appear in Duke Math Journal

References in corpus (4)

• Faber-Krahn inequalities in sharp quantitative form
• Some New Monotonicity Formulas and the Singular Set in the Lower Dimensional Obstacle Problem
• Direct epiperimetric inequalities for the thin obstacle problem and applications
• (Log-)epiperimetric inequality and regularity over smooth cones for almost Area-Minimizing currents

Cited by in corpus (1)

• Fractional Parabolic Theory as a High-Dimensional Limit of Fractional Elliptic Theory