  Given a class of nonautonomous elliptic operators $\A(t)$ with unbounded
coefficients, defined in $\overline{I \times \Om}$ (where $I$ is a
right-halfline or $I=\R$ and $\Om\subset \Rd$ is possibly unbounded), we prove
existence and uniqueness of the evolution operator associated to $\A(t)$ in the
space of bounded and continuous functions, under Dirichlet and first order, non
tangential homogeneous boundary conditions. Some qualitative properties of the
solutions, the compactness of the evolution operator and some uniform gradient
estimates are then proved.
