On Neumann and oblique derivatives boundary conditions for nonlocal elliptic equations |
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Authors: | Guy Barles Christine Georgelin Espen R. Jakobsen |
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Affiliation: | 1. Laboratoire de Mathématiques et Physique Théorique (UMR CNRS 7350), Fédération Denis Poisson (FR CNRS 2964), Université François-Rabelais Tours, Parc de Grandmont, 37200 Tours, France;2. Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491 Trondheim, Norway |
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Abstract: | Inspired by the penalization of the domain approach of Lions and Sznitman, we give a sense to Neumann and oblique derivatives boundary value problems for nonlocal, possibly degenerate elliptic equations. Two different cases are considered: (i) homogeneous Neumann boundary conditions in convex, possibly non-smooth and unbounded domains, and (ii) general oblique derivatives boundary conditions in smooth, bounded, and possibly non-convex domains. In each case we give appropriate definitions of viscosity solutions and prove uniqueness of solutions of the corresponding boundary value problems. We prove that these boundary value problems arise in the penalization of the domain limit from whole space problems and obtain as a corollary the existence of solutions of these problems. |
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Keywords: | Nonlocal elliptic equation Neumann-type boundary conditions General nonlocal operators Reflection Viscosity solutions Lé vy process |
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