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 $$Lu-\mathrm{div}(a(x)\nabla u(x))=f,\text{in }\hat{I}\text{c◯}\subset {\mathbb{R}}^n,$$ where $L$ is a general stable Lévy type operator and $a(\cdot )$ is a positive Hölder 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 $a(x)$ .