The Dirichlet problem for second-order elliptic integro-differential equations

In this talk, 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.