Higher H\"older regularity for mixed local and nonlocal degenerate elliptic equations

TL;DR

This paper proves higher regularity results, including almost Lipschitz continuity and explicit Hölder continuity, for solutions to equations involving combined local and nonlocal degenerate p-Laplace operators, along with existence and uniqueness results.

Contribution

It establishes new regularity results for mixed local and nonlocal degenerate elliptic equations, including explicit Hölder exponents and gradient continuity under certain conditions.

Findings

Almost Lipschitz regularity for homogeneous equations

Explicit Hölder continuity with computed exponents

Existence, uniqueness, and local boundedness of solutions

Abstract

We consider equations involving a combination of local and nonlocal degenerate $p$-Laplace operators. The main contribution of the paper is almost Lipschitz regularity for the homogeneous equation and H\"older continuity with an explicit H\"older exponent in the general case. For certain parameters, our results also imply H\"older continuity of the gradient. In addition, we establish existence, uniqueness and local boundedness. The approach is based on an iteration in the spirit of Moser combined with an approximation method.