$H^2$ regularity for the $p(x)-$Laplacian in two-dimensional convex domains

TL;DR

This paper investigates the global second-order regularity of solutions to the variable exponent p(x)-Laplacian in two-dimensional convex domains with Dirichlet boundary conditions, under certain regularity assumptions on p.

Contribution

It establishes $H^2$ regularity results for solutions of the p(x)-Laplacian in convex domains, extending regularity theory to variable exponent problems.

Findings

Proves $H^2$ regularity for solutions in convex domains.

Extends regularity results to variable exponent p(x)-Laplacian.

Provides conditions on p for regularity to hold.

Abstract

In this paper we study the $H^2$ global regularity for solutions of the $p(x)-$Laplacian in two dimensional convex domains with Dirichlet boundary conditions. Here $p:\Omega \to [p_1,\infty)$ with $p\in Lip(\bar{\Omega})$ and $p_1>1$.