On the Dirichlet problem for second-order elliptic integro-differential equations

In this article, we consider the analogue of the Dirichlet problem for second-order elliptic integro-differential equations, which consists in imposing the "boundary conditions" in the whole complementary of the domain. We are looking for conditions on the differential and integral parts of the equation in order to ensure that the Dirichlet boundary condition is satisfied in the classical sense or, in other words, in order that the solution agrees with the Dirichlet data on the boundary of the domain. We also provide a general existence result of a continuous viscosity solution of the non-local Dirichlet problem by using Perron's method.