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

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