Regularity of the free boundary for the obstacle problem for the fractional Laplacian with drift

TL;DR

This paper proves the $C^{1+eta}$ regularity of the free boundary in a fractional Laplacian obstacle problem with drift, using new monotonicity and epiperimetric inequalities.

Contribution

It introduces generalized monotonicity and epiperimetric inequalities for the fractional Laplacian with drift, extending classical methods to this nonlocal setting.

Findings

Established $C^{1+eta}$ regularity of the free boundary

Developed new monotonicity formula for fractional operators

Proved an epiperimetric inequality in this framework

Abstract

We establish the $C^{1+\gamma}$-H\"older regularity of the regular free boundary in the stationary obstacle problem defined by the fractional Laplace operator with drift in the subcritical regime. Our method of the proof consists in proving a new monotonicity formula and an epiperimetric inequality. Both tools generalizes the original ideas of G. Weiss for the classical obstacle problem to the framework of fractional powers of the Laplace operator with drift. Our study continues the earlier research, where two of us established the optimal interior regularity of solutions.