Mapping theorems for Sobolev spaces of vector-valued functions

We consider Sobolev spaces with values in Banach spaces, with emphasis on mapping properties. Our main results are the following: Given two Banach spaces X not equal{0} and Y, each Lipschitz continuous mapping F : X -> Y gives rise to a mapping u bar right arrow o u W-1,W- p (Omega, X) to W-1,W- p (Omega, Y) if and only if Y has the Radon Nikodym Property. But if in addition F is one-sided Gateaux differentiable, no condition on the space is needed. We also study when weak properties in the sense of duality imply strong properties. Our results are applied to prove embedding theorems, a multi-dimensional version of the Aubin Lions Lemma and characterizations of the space W-0(1, p) (Omega, X).