Equivalence of weak and viscosity solutions to mixed local and nonlocal $p$-Laplace equation

Abstract

In this paper, we establish the equivalence of weak and viscosity solutions for a homogeneous problem involving a mixed local and nonlocal elliptic operator in a bounded domain $\Omega \subset {\mathbb{R}}^N$ with Lipschitz boundary. We employ a comparison principle and a priori variational estimates to prove that continuous weak solutions are viscosity solutions and bounded viscosity solutions that vanish outside $\Omega$ are weak solutions. Our results are novel and new for mixed local and nonlocal operators, even for $p=2$.