Mixed local-nonlocal operators: maximum principles, eigenvalue problems and their applications

Abstract

In this article we consider a class of non-degenerate elliptic operators obtained by superpositioning the Laplacian and a general nonlocal operator. We study the existence-uniqueness results for Dirichlet boundary value problems, maximum principles and generalized eigenvalue problems. As applications to these results, we obtain Faber–Krahn inequality and a one-dimensional symmetry result related to the Gibbons’ conjecture. The latter results substantially extend the recent results of Biagi et al. [12, 10] who consider the operators of the form −Δ + (−Δ)s with s ∈ (0, 1).