Regularity theory for mixed local-nonlocal problem involving general stable operators

Abstract

In this paper, we study the regularity of solutions to a linear elliptic equation involving a mixed local-nonlocal operator of the form $$Lu - \operatorname{div}\big(a(x)\nabla u(x)\big)= f, \quad \text{in } \Omega \subset \mathbb{R}^n,$$ where $L$ is a general stable L\'{e}vy type operator and $a(\cdot)$ is a positive H\"{o}lder continuous weight. By establishing a maximum principle and a Liouville-type result in the entire space, we are able to derive the interior regularity and the regularity up to the boundary of the solutions under suitable assumptions on $f(x)$ and $a(x)$ .