Maximum principles, sliding techniques and applications to nonlocal equations

Abstract

This paper is devoted to the study of maximum principles holding for some nonlocal diffusion operators defined in (half-) bounded domains and its applications to obtain qualitative behaviors of solutions of some nonlinear problems. It is shown that, as in the classical case, the nonlocal diffusion considered satisfies a weak and a strong maximum principle. Uniqueness and monotonicity of solutions of nonlinear equations are therefore expected as in the classical case. It is first presented a simple proof of this qualitative behavior and the weak/strong maximum principle. An optimal condition to have a strong maximum for operator $mathcal{M}[u] :=Jstar u -u$ is also obtained. The proofs of the uniqueness and monotonicity essentially rely on the sliding method and the strong maximum principle.

Keywords

• Nonlocal diffusion operators
• maximum principles
• sliding methods.