Harnack inequality for nonlocal problems with non-standard growth

TL;DR

This paper establishes a comprehensive Harnack inequality for solutions to nonlocal problems exhibiting non-standard growth, advancing the understanding of regularity in such complex differential equations.

Contribution

It introduces a full Harnack inequality for nonlocal problems with non-standard growth, extending previous regularity results and providing new auxiliary bounds.

Findings

Proves local boundedness of solutions

Establishes a weak Harnack inequality for the De Giorgi class

Extends regularity theory to nonlocal problems with non-standard growth

Abstract

We prove a full Harnack inequality for local minimizers, as well as weak solutions to nonlocal problems with non-standard growth. The main auxiliary results are local boundedness and a weak Harnack inequality for functions in a corresponding De Giorgi class. This paper builds upon a recent work on regularity estimates for such nonlocal problems by the same authors.