Global gradient regularity and a Hopf lemma for quasilinear operators of mixed local-nonlocal type

TL;DR

This paper proves boundary regularity and a Hopf lemma for solutions of mixed local-nonlocal quasilinear equations, extending classical results to more complex operators with minimal boundary regularity assumptions.

Contribution

It introduces new regularity and boundary behavior results for mixed local-nonlocal quasilinear operators, including a Hopf lemma under weak boundary smoothness.

Findings

Solutions are $C^{1, heta}$ up to the boundary under certain conditions.

A Hopf lemma for positive supersolutions is established.

Results hold with minimal boundary regularity assumptions.

Abstract

We address some regularity issues for mixed local-nonlocal quasilinear operators modeled upon the sum of a $p$-Laplacian and of a fractional $(s, q)$-Laplacian. Under suitable assumptions on the right-hand sides and the outer data, we show that weak solutions of the Dirichlet problem are $C^{1, \theta}$-regular up to the boundary. In addition, we establish a Hopf type lemma for positive supersolutions. Both results hold assuming the boundary of the reference domain to be merely of class $C^{1, \alpha}$, while for the regularity result we also require that $p > s q$.