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 is also obtained. The proofs of the uniqueness and monotonicity essentially rely on the sliding method and the strong maximum principle.
Submitted August 7, 2006. Published May 10, 2007.
Math Subject Classifications: 35B50, 47G20, 35J60.
Key Words: Nonlocal diffusion operators; maximum principles; sliding methods.
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| Jérôme Coville |
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