Principal eigenvalue, maximum principles and linear stability for nonlocal diffusion equations in metric measure spaces

Abstract

We study principal eigenvalues and maximum principles for stationary nonlocal operators in spaces of integrable functions defined on general metric measure spaces under minimal assumptions on the kernels. Several characterizations of the principal eigenvalue are given as well as several conditions guaranteeing existence. Characterization of the (strong) maximum principle is also given. For evolution problems we prove the strong maximum principle and characterize stability in terms of the sign of the principal eigenvalue. We recover, extend and improve all previously known results, obtained for smooth open sets in euclidean space under continuity assumptions on the data.