Global regularity for the free boundary in the obstacle problem for the fractional Laplacian

TL;DR

This paper proves detailed regularity properties of the free boundary in the obstacle problem involving the fractional Laplacian, extending classical results to nonlocal operators and establishing new structural insights.

Contribution

It provides a comprehensive description of the free boundary's structure for the fractional Laplacian obstacle problem, including regular and singular points, which was previously known only for classical Laplacian cases.

Findings

Regular points form a $C^{1,eta}$ manifold

Singular points lie in unions of $C^1$ submanifolds

Established a new non-degeneracy condition at free boundary points

Abstract

We study the regularity of the free boundary in the obstacle problem for the fractional Laplacian under the assumption that the obstacle $\varphi$ satisfies $\Delta \varphi\leq 0$ near the contact region. Our main result establishes that the free boundary consists of a set of regular points, which is known to be a $(n-1)$-dimensional $C^{1,\alpha}$ manifold by the results in \cite{CSS}, and a set of singular points, which we prove to be contained in a union of $k$-dimensional $C^1$-submanifold, $k=0,\ldots,n-1$. Such a complete result on the structure of the free boundary was known only in the case of the classical Laplacian \cite{C-obst1,C-obst2}, and it is new even for the Signorini problem (which corresponds to the particular case of the $\frac12$-fractional Laplacian). A key ingredient behind our results is the validity of a new non-degeneracy condition $\sup_{B_r(x_0)}(u-\varphi)\geq c\,r^2$, valid at all free boundary points $x_0$.