Some remarks on stability of cones for the one-phase free boundary problem

TL;DR

This paper proves that stable cones in the one-phase free boundary problem are hyperplanes in four dimensions, leading to smoothness results for energy minimizing hypersurfaces in that dimension.

Contribution

It establishes that stable cones are hyperplanes in dimension four, providing new insights into the regularity of free boundary problems.

Findings

Stable cones are hyperplanes in dimension 4.

Energy minimizing hypersurfaces are smooth in dimension 4.

Results extend understanding of free boundary regularity.

Abstract

We show that stable cones for the one-phase free boundary problem are hyperplanes in dimension $4$. As a corollary, both one and two-phase energy minimizing hypersurfaces are smooth in dimension $4$.