Nondegeneracy for stable solutions to the one-phase free boundary problem

TL;DR

This paper proves a nondegeneracy condition for stable solutions to the one-phase free boundary problem using De Giorgi iteration and applies it to derive curvature bounds for stable free boundaries, especially in two dimensions.

Contribution

It establishes the nondegeneracy condition for stable solutions and links it to curvature bounds, advancing understanding of free boundary regularity.

Findings

Nondegeneracy condition proven for stable solutions.

Curvature bounds obtained for stable free boundaries in 2D.

Uses Sobolev inequality and mean curvature estimates.

Abstract

We prove the nondegeneracy condition for stable solutions to the one-phase free boundary problem. The proof is by a De Giorgi iteration, where we need the Sobolev inequality of Michael and Simon and, consequently, an integral estimate for the mean curvature of the free boundary. We then apply the nondegeneracy estimate to obtain local curvature bounds for stable free boundaries in dimension $n$, provided the Bernstein type theorem for stable, entire solutions in the same dimension is valid. In particular, we obtain this curvature estimate in $n=2$ dimensions.