The Dirichlet Problem for L\'evy-stable operators with $L^2$-data

TL;DR

This paper establishes Sobolev regularity for solutions to the Dirichlet problem involving Lévy-stable operators, including the fractional Laplacian, with rough exterior data in weighted $L^2$-spaces, bridging nonlocal and local theories.

Contribution

It proves Sobolev regularity for distributional solutions to Lévy-stable operators with rough exterior data, extending the theory to nonlocal operators and their local limits.

Findings

Sobolev regularity for Lévy-stable operators established

Robust estimates as $s o 1^-$ recover local theory

Applicable to fractional Laplacian and similar operators

Abstract

We prove Sobolev regularity for distributional solutions to the Dirichlet problem for generators of $2s$-stable processes and exterior data, inhomogeneity in weighted $L^2$-spaces. This class of operators includes the fractional Laplacian. For these rough exterior data the theory of weak variational solutions is not applicable. Our regularity estimate is robust in the limit $s\to 1-$ which allows us to recover the local theory.