Optimal regularity for nonlocal elliptic equations and free boundary problems

TL;DR

This paper proves the boundary regularity of solutions to nonlocal elliptic equations with general kernels and establishes optimal regularity results for obstacle problems, solving longstanding open questions in the field.

Contribution

It demonstrates for the first time $C^s$ boundary regularity for nonlocal elliptic equations with inhomogeneous kernels and proves optimal $C^{1+s}$ regularity for obstacle problems, extending previous homogeneous kernel results.

Findings

Boundary regularity holds for general kernels, not just homogeneous.

Optimal $C^{1+s}$ regularity established for obstacle problems.

Constructs 1D solutions as minimizers to analyze free boundary problems.

Abstract

In this article we establish for the first time the $C^s$ boundary regularity of solutions to nonlocal elliptic equations with kernels $K(y)\asymp |y|^{-n-2s}$. This was known to hold only when $K$ is homogeneous, and it is quite surprising that it holds for general inhomogeneous kernels, too. As an application of our results, we also establish the optimal $C^{1+s}$ regularity of solutions to obstacle problems for general nonlocal operators with kernels $K(y)\asymp |y|^{-n-2s}$. Again, this was only known when $K$ is homogeneous, and it solves a long-standing open question in the field. A new key idea is to construct a 1D solution as a minimizer of an appropriate nonlocal one-phase free boundary problem, for which we establish optimal $C^s$ regularity and non-degeneracy estimates.