Sobolev regularity for the nonlocal $(1, p)$-Laplace equations in the superquadratic case

TL;DR

This paper establishes interior Sobolev regularity and Hölder continuity for solutions to nonlocal (1,p)-Laplace equations in the superquadratic case, using advanced analytical techniques.

Contribution

It provides the first detailed analysis of regularity for nonlocal (1,p)-Laplace equations in the superquadratic regime, including explicit Hölder estimates.

Findings

Proved interior Sobolev regularity of solutions.

Derived explicit Hölder continuity estimates.

Developed new analytical techniques for nonlocal equations.

Abstract

We investigate the interior Sobolev regularity of weak solutions to the nonlocal $(1, p)$-Laplace equations in the superquadratic case $p\ge 2$. As a product, the explicit H\"{o}lder continuity estimates of weak solutions are derived. The proof relies on a detailed analysis of the structural characteristics of $(1, p)$-growth in the nonlocal setting, combined with the finite difference quotient method, tail estimates, refined energy estimates, and a Moser-type iteration scheme.