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

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 .