Higher regularity in nonlocal free boundary problems

TL;DR

This paper proves that free boundaries in nonlocal problems of order 2s are infinitely smooth under certain conditions, extending known results for the fractional Laplacian and introducing new techniques for boundary regularity.

Contribution

It establishes higher regularity of free boundaries in nonlocal problems, including the fractional Laplacian, and develops new tools like integration by parts formulas and boundary estimates.

Findings

C^{2,eta} free boundaries are C^ for the nonlocal Bernoulli problem.

New proof of higher regularity for free boundaries in the nonlocal obstacle problem.

Development of novel boundary Hlder estimates and integration by parts formulas for nonlocal equations.

Abstract

We study the higher regularity in nonlocal free boundary problems posed for general integro-differential operators of order $2s$. Our main result is for the nonlocal one-phase (Bernoulli) problem, for which we establish that $C^{2,\alpha}$ free boundaries are $C^\infty$. This is new even for the fractional Laplacian, as it was only known in case $s=\frac12$. We also establish a general result for overdetermined problems, showing that if the boundary condition is smooth, then so is $\partial\Omega$. Our approach is very robust and works as well for the nonlocal obstacle problem, where it yields a new proof of the higher regularity of free boundaries, completely different from the one in [AbRo20]. In order to prove our results, we need to develop, among other tools, new integration by parts formulas and delicate boundary H\"older estimates for nonlocal equations with (local) Neumann boundary conditions that had not been studied before and are of independent interest.