Abstract

In this paper we are interested in functions defined, on a set of matrices, by the mean of quadratic forms and we compute the rank-one-convex, quasiconvex, polyconvex and convex envelopes of these functions. For that, and for a given quadratic form, we prove, in a first part, some general decomposition results for matrices, with a rank-one-compatibility condition. We also study the James-Ericksen stored energy function.

Author information

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M. Bousselsal

Département de Mathématiques, Ecole Normale Supérieure Vieux Kouba 16050 Algiers, Algeria.

B. Brighi

Centre d’Analyse Non Linéaire, Département de Mathématiques, URA-CNRS 399, Université de Metz, Ile du Saulcy, 57045 Metz Cedex 01, France

Keywords

  • rank-one-convex
  • quasiconvex
  • envelope
  • quadratic form
  • James-Ericksen function
  • Pipkin’s formula

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Suggested citation

M. Bousselsal, B. Brighi. “Rank-One-Convex and Quasiconvex Envelopes for Functions Depending on Quadratic Forms.” Journal of Convex Analysis 4 (1997), No. 2, 305–319. https://doi.org/10.68381/jca04016

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Published by Heldermann Verlag, 1997. Rights now held by Banach Press.