A Dirichlet problem for nonlocal degenerate elliptic operators with internal nonlinearity

TL;DR

This paper investigates a Dirichlet problem involving nonlocal degenerate elliptic operators with internal nonlinearities, establishing existence, uniqueness, and regularity of solutions under various conditions.

Contribution

It introduces new existence and regularity results for solutions to nonlocal degenerate elliptic equations with internal nonlinearities, including classical and smooth solutions.

Findings

Existence and uniqueness of viscosity solutions under mild boundary conditions

Classical solutions when the operator is uniformly elliptic

Smooth solutions if the operator and boundary data are smooth

Abstract

We study a Dirichlet problem in the entire space for some nonlocal degenerate elliptic operators with internal nonlinearities. With very mild assumptions on the boundary datum, we prove existence and uniqueness of the solution in the viscosity sense. If we further assume uniform ellipticity then the solution is shown to be classical, and even smooth if both the operator and the boundary datum are smooth.