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

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.