Electron. J. Diff. Eqns., Vol. 2007(2007), No. 68, pp. 1-23.

Maximum principles, sliding techniques and applications to nonlocal equations

Jerome Coville

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 $\mathcal{M}[u] :=J\star u -u$ 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
Laboratoire CEREMADE, Université Paris Dauphine
Place du Maréchal De Lattre De Tassigny
75775 Paris Cedex 16, France.
Centro de Modelamiento Matemático
UMI 2807 CNRS-Universidad de Chile
Blanco Encalada 2120 - 7 Piso
Casilla 170 - Correo 3, Santiago, Chile
email: coville@dim.uchile.cl

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