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On Neumann type problems for nonlocal equations set in a half space


Authors: Guy Barles, Emmanuel Chasseigne, Christine Georgelin and Espen R. Jakobsen
Journal: Trans. Amer. Math. Soc. 366 (2014), 4873-4917
MSC (2010): Primary 35R09; Secondary 45K05, 35D40
DOI: https://doi.org/10.1090/S0002-9947-2014-06181-3
Published electronically: March 5, 2014
MathSciNet review: 3217703
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Abstract: We study Neumann type boundary value problems for nonlocal equations related to Lévy type processes. Since these equations are nonlocal, Neumann type problems can be obtained in many ways, depending on the kind of ``reflection'' we impose on the outside jumps. To focus on the new phenomena and ideas, we consider different models of reflection and rather general nonsymmetric Lévy measures, but only simple linear equations in half-space domains. We derive the Neumann/reflection problems through a truncation procedure on the Lévy measure, and then we develop a viscosity solution theory which includes comparison, existence, and some regularity results. For problems involving fractional Laplace $ (-\Delta )^{\frac \alpha 2}$-like nonlocal operators, we prove that solutions of all our nonlocal Neumann problems converge as $ \alpha \rightarrow , 2^-$ to the solution of a classical local Neumann problem. The reflection models we consider include cases where the underlying Lévy processes are reflected, projected, and/or censored when exiting the domain.


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Additional Information

Guy Barles
Affiliation: Laboratoire de Mathématiques et Physique Théorique, CNRS UMR 7350, Fédération Denis Poisson, FR CNRS 2964, Université François Rabelais, Parc de Grandmont, 37200 Tours, France
Email: barles@lmpt.univ-tours.fr

Emmanuel Chasseigne
Affiliation: Laboratoire de Mathématiques et Physique Théorique, CNRS UMR 7350, Fédération Denis Poisson, FR CNRS 2964, Université François Rabelais, Parc de Grandmont, 37200 Tours, France
Email: Emmanuel.Chasseigne@lmpt.univ-tours.fr

Christine Georgelin
Affiliation: Laboratoire de Mathématiques et Physique Théorique, CNRS UMR 7350, Fédération Denis Poisson, FR CNRS 2964, Université François Rabelais, Parc de Grandmont, 37200 Tours, France
Email: christine.georgelin@lmpt.univ-tours.fr

Espen R. Jakobsen
Affiliation: Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491 Trondheim, Norway
Email: erj@math.ntnu.no

DOI: https://doi.org/10.1090/S0002-9947-2014-06181-3
Keywords: Nonlocal equations, Neumann boundary conditions, jumps, L\'evy measure, reflection, viscosity solutions
Received by editor(s): December 2, 2011
Received by editor(s) in revised form: January 17, 2013
Published electronically: March 5, 2014
Additional Notes: The fourth author was supported by the Research Council of Norway through the project “DIMMA”
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.