Regularity of higher order in two-phase free boundary problems

TL;DR

This paper advances the understanding of two-phase free boundary problems by proving that flat or Lipschitz free boundaries are locally twice differentiable with Hölder continuous second derivatives.

Contribution

It extends previous methods to establish higher regularity ($C^{2,eta}$) of free boundaries in inhomogeneous two-phase problems.

Findings

Flat or Lipschitz free boundaries are locally $C^{2,eta}$.

The regularity results apply to inhomogeneous two-phase free boundary problems.

The approach builds on and extends previous strategies for boundary regularity.

Abstract

We develop further the strategy implemented in our series of papers on inhomogeneous two-phase fee boundary problems, to show that flat or Lipschitz free boundaries of such problems are locally $C^{2,\gamma }.$