On Neumann Type Problems for nonlocal Equations set in a half Space
By Guy Barles, Emmanuel Chasseigne, C. Georgelin, Espen Robstad Jakobsen. 2011
60 citations recorded by the source corpus.
We study Neumann type boundary value problems for nonlocal equations related to Levy 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 phenomenas and ideas, we consider different models of reflection and rather general non-symmetric Levy measures, but only simple linear equations in half-space domains. We derive the Neumann/reflection problems through a truncation procedure on the Levy measure, and then we develop a viscosity solution theory which includes comparison, existence, and some regularity results. For problems involving fractional Laplacian type operators like e.g.$(-\Delta)^{\alpha/2}$, we prove that solutions of all our nonlocal Neumann problems converge as alpha goes to 2 to the solution of a classical Neumann problem. The reflection models we consider include cases where the underlying Levy processes are reflected, projected, and/or censored upon exiting the domain.
Subfields: Applied Mathematics, Computational Theory and Mathematics
View Paper: https://www.semanticscholar.org/paper/6000f38cabb8d277b103320cb5475b0342403fac
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In Computational Theory and Mathematics
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- G. Em Karniadakis
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- Xiaoning Zheng
- Mark Ainsworth
Citations in the corpus
- What is the fractional Laplacian? A comparative review with new results (2018, background): “Firstly, because the formulation of the nonlocal Neumann condition is a subject of some controversy, as there are multiple formulations in the literature with no consensus [66,20,112,21,113].”
- Nonlocal problems with Neumann boundary conditions (2014, background): “In [1, 3] the idea was to find a deterministic “reflection” or “projection” which describes the way in which the particle comes back into Ω.”