Two-phase almost minimizers for a fractional free boundary problem

TL;DR

This paper investigates the regularity and structure of free boundaries in a fractional free boundary problem, establishing that positive and negative free boundaries do not touch and that flatness implies smoothness.

Contribution

It proves the non-contact of positive and negative free boundaries and establishes regularity results, including $C^{0,s}$ regularity and $C^{1,eta}$ smoothness under flatness conditions.

Findings

Positive and negative free boundaries do not touch.

Almost minimizers are $C^{0,s}$ regular.

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

Abstract

In this paper, we study almost minimizers to a fractional Alt-Caffarelli-Friedman type functional. Our main results concern the optimal $C^{0,s}$ regularity of almost minimizers as well as the structure of the free boundary. We first prove that the two free boundaries $F^+(u)=\partial\{u(\cdot,0)>0\}$ and $F^-(u)=\partial\{u(\cdot,0)<0\}$ cannot touch, that is, $F^+(u)\cap F^-(u)=\emptyset$. Lastly, we prove a flatness implies $C^{1,\gamma}$ result for the free boundary.