Gradient estimates for parabolic problems with unbounded coefficients in non convex unbounded domains

We consider a class of uniformly elliptic operators A with unbounded coefficients in unbounded domains Omega. Under suitable assumptions on the geometry of Omega and on the coefficients, we prove that the Cauchy-Neumann problem associated with the operator A admits a unique bounded classical solution u for any initial datum f which is bounded and continuous in (Omega) over bar. Moreover, we prove uniform and pointwise gradient estimates for u. Finally, we give some applications of the so obtained estimates.