Boundary regularity and Hopf lemma for nondegenerate stable operators

TL;DR

This paper establishes sharp boundary regularity and a Hopf lemma for solutions to stable integro-differential equations, including the fractional Laplacian, with results that smoothly connect to classical second-order equations as the order approaches 1.

Contribution

It provides the first sharp boundary Hölder regularity results for stable operators in domains with $C^{1, ext{dini}}$ exterior regularity, extending classical PDE results to nonlocal operators.

Findings

Sharp boundary Hölder regularity for stable operators.

A Hopf-type boundary lemma for these operators.

Results are robust as the order approaches 1, recovering classical PDE results.

Abstract

We prove sharp boundary H{\"o}lder regularity for solutions to equations involving stable integro-differential operators in bounded open sets satisfying the exterior $C^{1,\text{dini}}$-property. This result is new even for the fractional Laplacian. A Hopf-type boundary lemma is proven, too. An additional feature of this work is that the regularity estimate is robust as $s\to 1-$ and we recover the classical results for second order equations.