Improvement of flatness for nonlocal free boundary problems

TL;DR

This paper proves that the free boundary in nonlocal one-phase problems is smooth near regular points and classifies blow-up limits in 2D, extending known results from fractional Laplacian to general nonlocal operators.

Contribution

It introduces a new flatness improvement scheme for nonlocal free boundary problems, establishing regularity results for general nonlocal operators, not just fractional Laplacians.

Findings

Free boundary is $C^{1,\alpha}$ near regular points.

Set of regular points is open and dense.

Complete classification of blow-up limits in 2D.

Abstract

In this article we study for the first time the regularity of the free boundary in the one-phase free boundary problem driven by a general nonlocal operator. Our main results establish that the free boundary is $C^{1,\alpha}$ near regular points, and that the set of regular free boundary points is open and dense. Moreover, in 2D we classify all blow-up limits and prove that the free boundary is $C^{1,\alpha}$ everywhere. The main technical tool of our proof is an improvement of flatness scheme, which we establish in the general framework of viscosity solutions, and which is of independent interest. All of these results were only known for the fractional Laplacian, and are completely new for general nonlocal operators. In contrast to previous works on the fractional Laplacian, our method of proof is purely nonlocal in nature.