Journal of Convex Analysis, Vol. 4, No. 2, pp. 305-319 (1997)

Rank-one-convex and Quasiconvex Envelopes for Functions Depending on Quadratic Forms

M. Bousselsal and B. Brighi

Département de Mathématiques, Ecole Normale Supérieure, Vieux Kouba 16050 Algiers, Algeria, and Centre d'Analyse Non Linéaire, Département de Mathématiques, URA-CNRS 399, Université de Metz, Ile du Saulcy, 57045 METZ Cedex 01, France, brighi@poncelet.univ-metz.fr

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.

Keywords: rank-one-convex, quasiconvex, envelope, quadratic form, James-Ericksen function, Pipkin's formula

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