Existence and Uniqueness for Integro-Differential Equations with Dominating Drift Terms

TL;DR

This paper establishes the existence and uniqueness of viscosity solutions for certain integro-differential elliptic equations with dominating drift terms, addressing boundary condition issues and providing comparison principles.

Contribution

It introduces a framework for well-posedness of Dirichlet problems with lower-order elliptic operators and general boundary conditions, extending previous results.

Findings

Proved strong comparison principles under general drift assumptions

Established existence and uniqueness of solutions in continuous function spaces

Addressed boundary condition loss issues for non-diffusive operators

Abstract

In this paper we are interested on the well-posedness of Dirichlet problems associated to integro-differential elliptic operators of order $\alpha < 1$ in a bounded smooth domain $\Omega$ . The main difficulty arises because of losses of the boundary condition for sub and supersolutions due to the lower diffusive effect of the elliptic operator compared with the drift term. We consider the notion of viscosity solution with generalized boundary conditions, concluding strong comparison principles in $\bar{\Omega}$ under rather general assumptions over the drift term. As a consequence, existence and uniqueness of solutions in $C(\bar{\Omega})$ is obtained via Perron's method.