Homotopy types for tamely approximable Sobolev maps between manifolds

We define a class ^p (M,N) of Sobolev maps from a manifoldM into a manifoldN, in such a way that each mapu ^p (M, N) has a well defined p]-homotopy type, providedN satisfies a topological hypothesis. Using this, we prove the existence of minimizers in p]-homotopy classes for some polyconvex variational problems.