Free boundary regularity for a problem with right hand side

TL;DR

This paper establishes regularity results for free boundaries in a one-phase problem with variable coefficients and a non-zero right hand side, using a novel approach to prove $C^{1,eta}$ regularity.

Contribution

It introduces a new method to prove $C^{1,eta}$ regularity for free boundaries, extending classical results to more general settings.

Findings

Flat free boundaries are $C^{1,eta}$.

Lipschitz free boundaries are $C^{1,eta}$.

The approach differs from classical supconvolution methods.

Abstract

We consider a one-phase free boundary problem with variable coefficients and non-zero right hand side. We prove that flat free boundaries are $C^{1,\alpha}$ using a different approach than the classical supconvolution method of Caffarelli. We use this result to obtain that Lipschitz free boundaries are $C^{1,\alpha}$.