Regularity in a one-phase free boundary problem for the fractional Laplacian

TL;DR

This paper proves that flat free boundaries in a one-phase fractional Laplacian problem are smooth, extending classical Bernoulli free boundary regularity results to the fractional setting.

Contribution

It establishes $C^{1,eta}$ regularity of flat free boundaries for the fractional Laplacian, generalizing classical results to nonlocal operators.

Findings

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

Regularity results extend classical Bernoulli problem to fractional Laplacian.

Provides a foundation for further regularity studies in nonlocal free boundary problems.

Abstract

For a one-phase free boundary problem involving a fractional Laplacian, we prove that "flat free boundaries" are $C^{1,\alpha}$. We recover the regularity results of Caffarelli for viscosity solutions of the classical Bernoulli-type free boundary problem with the standard Laplacian.