Generic regularity of free boundaries in the obstacle problem for the fractional Laplacian

TL;DR

This paper proves that for almost every obstacle, the free boundary in the fractional obstacle problem is regular up to dimension 3 for all s in (0,1), extending previous results to a broader fractional setting.

Contribution

It extends regularity results of free boundaries to the fractional Laplacian for all s in (0,1) and general obstacles, including new frequency gap results.

Findings

Free boundary contains only regular points up to dimension 3 for almost every obstacle.

Extended fine structure analysis of free boundaries to fractional Laplacian with s in (0,1).

Established explicit uniform frequency gaps for solutions.

Abstract

We establish generic regularity results of free boundaries for solutions of the obstacle problem for the fractional Laplacian $(-\Delta)^s$. We prove that, for almost every obstacle, the free boundary contains only regular points up to dimension $3$, for every $s\in(0,1)$. To do so, we extend some results on the fine structure of the free boundary to the case $s\in (0,1)$ and general non-zero obstacle, including a blow-up analysis at points with frequency $2m+2s$, and we prove new explicit uniform frequency gaps for solutions of the fractional obstacle problem.