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

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 $$\smash {C^{1,\alpha }}$$ regularity for the viscosity solutions of the normalized p(x)-Laplace equation. As an additional application, we prove a Rado-type removability theorem.