Equivalence of Viscosity and Weak Solutions for the $p(x)$-Laplacian

TL;DR

This paper proves that viscosity and weak solutions to the $p(x)$-Laplace equation are equivalent, establishing uniqueness and a Radó-type removability theorem for solutions with variable exponent growth.

Contribution

It demonstrates the equivalence of viscosity and weak solutions for the $p(x)$-Laplace equation, a significant step in understanding solutions with variable exponents.

Findings

Viscosity supersolutions and $p(x)$-superharmonic functions coincide.

Weak and viscosity solutions are the same class of functions.

Viscosity solutions to Dirichlet problems are unique.

Abstract

We consider different notions of solutions to the $p(x)$-Laplace equation $-\div(\abs{Du(x)}^{p(x)-2}Du(x))=0$ with $ 1<p(x)<\infty$. We show by proving a comparison principle that viscosity supersolutions and $p(x)$-superharmonic functions of nonlinear potential theory coincide. This implies that weak and viscosity solutions are the same class of functions, and that viscosity solutions to Dirichlet problems are unique. As an application, we prove a Rad\'o type removability theorem.