On Neumann type problems for nonlocal equations set in a half space
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- by Guy Barles, Emmanuel Chasseigne, Christine Georgelin and Espen R. Jakobsen PDF
- Trans. Amer. Math. Soc. 366 (2014), 4873-4917 Request permission
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.References
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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
- 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”
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3217703