On Neumann Type Problems for nonlocal Equations set in a half Space

TL;DR

This paper develops a viscosity solution framework for Neumann boundary problems involving nonlocal operators related to Lévy processes, analyzing different reflection models and showing convergence to classical solutions as the fractional order approaches 2.

Contribution

It introduces a new approach to nonlocal Neumann problems with general Lévy measures, including multiple reflection models, and proves convergence to classical Neumann solutions.

Findings

Established comparison, existence, and regularity results for nonlocal Neumann problems.

Derived convergence of solutions to classical Neumann problems as fractional order approaches 2.

Analyzed various reflection models for Lévy processes in half-space domains.

Abstract

We study Neumann type boundary value problems for nonlocal equations related to L\'evy 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 L\'evy measures, but only simple linear equations in half-space domains. We derive the Neumann/reflection problems through a truncation procedure on the L\'evy 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 L\'evy processes are reflected, projected, and/or censored upon exiting the domain.