Regularity Results of the Thin Obstacle Problem for the $p(x)$-Laplacian

TL;DR

This paper investigates the regularity properties of solutions to the thin obstacle problem involving the $p(x)$-Laplacian, establishing higher integrability and Hölder continuity of the gradient under certain conditions.

Contribution

It provides new regularity results for the thin obstacle problem with variable exponent growth, extending classical results to the $p(x)$-Laplacian case.

Findings

Proved higher integrability of the gradient of minimizers.

Established Hölder regularity of the gradient.

Extended regularity theory to variable exponent settings.

Abstract

We study thin obstacle problems involving the energy functional with $p(x)$-growth. We prove higher integrability and H\"{o}lder regularity for the gradient of minimizers of the thin obstacle problems under the assumption that the variable exponent $p(x)$ is H\"{o}lder continuous.