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

Abstract

We consider different notions of solutions to the $p(x)$-Laplace equation $-÷(\abs{Du(x)}^{p(x)-2}Du(x))=0$ 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 Radó type removability theorem.

References in corpus (2)

• Tug-of-war and the infinity Laplacian
• p(x)-Harmonic functions with unbounded exponent in a subdomain

Cited by in corpus (4)

• Generalized Harnack's inequality for nonhomogeneous elliptic equations
• Regularity of flat free boundaries for a $p(x)$-Laplacian problem with right hand side
• Regularity of Lipschitz free boundaries for a $p(x)$-Laplacian problem with right hand side
• A Carleson type inequality for fully nonlinear elliptic equations with non-Lipschitz drift term