Original article

Calculus of Variations and Partial Differential Equations

, Volume 11, Issue 4, pp 333-359

Regularity of quasiconvex envelopes

  • John M. BallAffiliated withMathematical Institute, University of Oxford, OX1 3LB Oxford, England
  • , Bernd KirchheimAffiliated withMax-Planck-Institut für Mathematik in den Naturwissenschaften, Inselstr. 22–26, 04103 Leipzig, Germany
  • , Jan KristensenAffiliated withMathematical Institute, University of Oxford, OX1 3LB Oxford, England

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Abstract.

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.