Regularity of quasiconvex envelopes

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We prove that the quasiconvex envelope of a differentiable function which satisfies natural growth conditions at infinity is a $C^1$ function. Without the growth conditions the result fails in general. We also obtain results on higher regularity (in the sense of $C^{1,\alpha}_{\rm loc}$ ) and similar results for other types of envelopes, including polyconvex and rank-1 convex envelopes.

Received January 11, 2000/ Accepted January 14, 2000 / Published online June 28, 2000