Equivalence of viscosity and weak solutions for the normalized $p(x)$-Laplacian

TL;DR

This paper proves the equivalence between viscosity and weak solutions for the normalized p(x)-Laplacian, establishing regularity results and a removability theorem, advancing understanding of variable exponent PDEs.

Contribution

It demonstrates the equivalence of viscosity and weak solutions for the normalized p(x)-Laplacian under Lipschitz conditions on p, and derives regularity and removability results.

Findings

Viscosity solutions coincide with weak solutions for the normalized p(x)-Laplacian.

Established C^{1,α} regularity for viscosity solutions.

Proved a Radó-type removability theorem.

Abstract

We show that viscosity solutions to the normalized $p(x)$-Laplace equation coincide with distributional weak solutions to the strong $p(x)$-Laplace equation when $p$ is Lipschitz and $\inf p>1$. This yields $C^{1,\alpha}$ regularity for the viscosity solutions of the normalized $p(x)$-Laplace equation. As an additional application, we prove a Rad\'o-type removability theorem.