Optimal boundary regularity for mixed local and nonlocal equations

TL;DR

This paper establishes optimal boundary regularity estimates for solutions to elliptic equations combining local and nonlocal operators, using weighted Hölder spaces and fixed-point techniques.

Contribution

It provides the first sharp boundary regularity results for mixed local and nonlocal elliptic equations, including explicit counterexamples for optimality.

Findings

Sharp boundary regularity estimates derived

Counterexamples demonstrating optimality

Extension of regularity theory to mixed operators

Abstract

We provide sharp boundary regularity estimates for solutions to elliptic equations driven by an integro-differential operator obtained as the sum of a Laplacian with a nonlocal operator generalizing a fractional Laplacian. Our approach makes use of weighted H\"older spaces as well as regularity estimates for the Laplacian in this context and a fixed-point argument. We show the optimality of the obtained estimates by means of a counterexample that we have striven to keep as explicit as possible.