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

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.