Remarks on the quasiconvex envelope of some functions depending on quadratic forms

M. Bousselsal; H. Le Dret

Bollettino dell'Unione Matematica Italiana (2002)

  • Volume: 5-B, Issue: 2, page 469-486
  • ISSN: 0392-4041

Abstract

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We compute the quasiconvex envelope of certain functions defined on the space M m ⁢ n of real m × n matrices. These functions are basically functions of a quadratic form on M m ⁢ n . The quasiconvex envelope computation is applied to densities that are related to the James-Ericksen elastic stored energy function.

How to cite

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Bousselsal, M., and Le Dret, H.. "Remarks on the quasiconvex envelope of some functions depending on quadratic forms." Bollettino dell'Unione Matematica Italiana 5-B.2 (2002): 469-486. <http://eudml.org/doc/194950>.

@article{Bousselsal2002,
abstract = {We compute the quasiconvex envelope of certain functions defined on the space $M_\{mn\}$ of real $m \times n$ matrices. These functions are basically functions of a quadratic form on $M_\{mn\}$. The quasiconvex envelope computation is applied to densities that are related to the James-Ericksen elastic stored energy function.},
author = {Bousselsal, M., Le Dret, H.},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {6},
number = {2},
pages = {469-486},
publisher = {Unione Matematica Italiana},
title = {Remarks on the quasiconvex envelope of some functions depending on quadratic forms},
url = {http://eudml.org/doc/194950},
volume = {5-B},
year = {2002},
}

TY - JOUR
AU - Bousselsal, M.
AU - Le Dret, H.
TI - Remarks on the quasiconvex envelope of some functions depending on quadratic forms
JO - Bollettino dell'Unione Matematica Italiana
DA - 2002/6//
PB - Unione Matematica Italiana
VL - 5-B
IS - 2
SP - 469
EP - 486
AB - We compute the quasiconvex envelope of certain functions defined on the space $M_{mn}$ of real $m \times n$ matrices. These functions are basically functions of a quadratic form on $M_{mn}$. The quasiconvex envelope computation is applied to densities that are related to the James-Ericksen elastic stored energy function.
LA - eng
UR - http://eudml.org/doc/194950
ER -

References

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  4. COLLINS, C.- LUSKIN, M., Numerical modeling of the microstructure of crystals with symmetry-related variants, in Proceedings of the ARO US-Japan Workshop on Smart/Intelligent Materials and Systems, Honolulu, Hawaii, Technomic Publishing Company, Lancaster, PA, 1990. 
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  8. LE DRET, H.- RAOULT, A., The quasiconvex envelope of the Saint Venant-Kirchhoff stored energy function, Proc. Roy. Soc. Edinburgh A, 125 (1995), 1179-1192. Zbl0843.73016MR1362998
  9. LE DRET, H.- RAOULT, A., Quasiconvex envelopes of stored energy densities that are convex with respect to the strain tensor, in Calculus of Variations, Applications and Computations, Pont-a-Mousson 1994 (C. Bandle, J. Bemelmans, M. Chipot, J. Saint Jean Paulin, I. Shafrir eds.), 138-146, Pitman Research Notes in Mathematics, Longman, 1995. Zbl0830.73012MR1419340
  10. MORREY, C. B. JR., Quasiconvexity and the semicontinuity of multiple integrals, Pacific J. Math., 2 (1952), 25-53. Zbl0046.10803MR54865
  11. MORREY, C. B. JR., Multiple Integrals in the Calculus of Variations, Springer-Verlag, Berlin, 1966. Zbl0142.38701MR202511
  12. ŠVERÁK, V., Rank-one-convexity does not imply quasiconvexity, Proc. Royal Soc. Edinburgh A, 120 (1992), 185-189. Zbl0777.49015MR1149994

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