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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 M := 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
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Jerome Coville. Maximum principles, sliding techniques and applications to nonlocal equations. Electronic Journal of Differential Equations, Texas State University, Department of Mathematics, 2007, 2007 (68), 23 p. ⟨hal-02654682⟩



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