Regularity of quasiconvex envelopes

We prove that the quasiconvex envelope of a differentiable function which satisfies natural growth conditions at infinity is a C1 function. Without the growth conditions the result fails in general. We also obtain results on higher regularity (in the sense of C1,αloc) and similar results for other types of envelopes, including polyconvex and rank-1 convex envelopes.