Regularity for nonlocal problems with non-standard growth

TL;DR

This paper establishes local boundedness and H"older continuity for minimizers and solutions of nonlocal problems with non-standard growth, using De Giorgi class methods.

Contribution

It introduces regularity results for nonlocal functionals with (p,q)-growth, extending classical techniques to nonlocal and non-standard growth contexts.

Findings

Minimizers are locally bounded.

Weak solutions are H"older continuous.

Results apply to a broad class of nonlocal problems.

Abstract

We study robust regularity estimates for local minimizers of nonlocal functionals with non-standard growth of $(p,q)$-type and for weak solutions to a related class of nonlocal equations. The main results of this paper are local boundedness and H\"older continuity of minimizers and weak solutions. Our approach is based on the study of corresponding De Giorgi classes.