On Neumann and oblique derivatives boundary conditions for nonlocal elliptic equations

TL;DR

This paper develops a framework for defining and analyzing Neumann and oblique derivative boundary conditions for nonlocal elliptic equations, extending the theory to non-smooth, unbounded, and non-convex domains.

Contribution

It introduces new definitions of viscosity solutions for nonlocal boundary conditions and proves their uniqueness and existence in various domain settings.

Findings

Established well-posedness of boundary value problems for nonlocal elliptic equations.

Connected boundary problems to penalization limits from whole space problems.

Extended the theory to non-smooth and non-convex domains.

Abstract

Inspired by the penalization of the domain approach of Lions & Sznitman, we give a sense to Neumann and oblique derivatives boundary value problems for nonlocal, possibly degenerate elliptic equations. Two different cases are considered: (i) homogeneous Neumann boundary conditions in convex, possibly non-smooth and unbounded domains, and (ii) general oblique derivatives boundary conditions in smooth, bounded, and possibly non-convex domains. In each case we give apropriate definitions of viscosity solutions and prove uniqueness of solutions of the corresponding boundary value problems. We prove that these boundary value problems arise in the penalization of the domain limit from whole space problems and obtain as a corollary the existence of solutions of these problems.