Remarks on the quasiconvex envelope of some functions depending on quadratic forms
Bousselsal, M. ; Le Dret, H.
Bollettino dell'Unione Matematica Italiana, Tome 5-A (2002), p. 469-486 / Harvested from Biblioteca Digitale Italiana di Matematica
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We compute the quasiconvex envelope of certain functions defined on the space Mm⁢n of real m×n matrices. These functions are basically functions of a quadratic form on Mm⁢n. The quasiconvex envelope computation is applied to densities that are related to the James-Ericksen elastic stored energy function.

In questo lavoro calcoliamo la chiusura quasi convessa di alcune funzioni definite sullo spazio Mm⁢n delle matrici reali m×n attraverso forme quadratiche. I risultati sono applicati ad alcune funzioni relative alla densità di energia elastica di James e Ericksen.

Publié le : 2002-06-01
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     author = {M. Bousselsal and H. Le Dret},
     title = {Remarks on the quasiconvex envelope of some functions depending on quadratic forms},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {5-A},
     year = {2002},
     pages = {469-486},
     zbl = {1177.49009},
     mrnumber = {1911201},
     language = {en},
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Bousselsal, M.; Le Dret, H. Remarks on the quasiconvex envelope of some functions depending on quadratic forms. Bollettino dell'Unione Matematica Italiana, Tome 5-A (2002) pp. 469-486. http://gdmltest.u-ga.fr/item/BUMI_2002_8_5B_2_469_0/

[1] Ball, J. M., Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal., 64 (1977), 337-403. | MR 475169 | Zbl 0368.73040

[2] Bhattacharya, K.-Dolzman, G., Relaxation of some multi-well problems, to appear, 1998. | MR 1830413 | Zbl 0977.74029

[3] Bousselsal, M.-Brighi, B., Rank-one-convex and quasiconvex envelopes for functions depending on quadratic forms, J. Convex Anal., 4 (1997), 305-319. | MR 1613479 | Zbl 0903.49011

[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.

[5] Dacorogna, B., Direct Methods in the Calculus of Variations, Applied Mathematical Sciences, no. 78, Springer-Verlag, Berlin, 1989. | MR 990890 | Zbl 0703.49001

[6] Kohn, R. V.-Strang, G., Explicit relaxation of a variational problem in optimal design, Bull. A.M.S., 9 (1983), 211-214. | MR 707959 | Zbl 0527.49002

[7] Kohn, R. V.-Strang, G., Optimal design and relaxation of variational problems, I, II and III, Comm. Pure Appl. Math., 39 (1986), 113-137, 139-182 and 353-377. | MR 820342 | Zbl 0621.49008

[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. | MR 1362998 | Zbl 0843.73016

[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. | MR 1419340 | Zbl 0830.73012

[10] Morrey, C. B. Jr., Quasiconvexity and the semicontinuity of multiple integrals, Pacific J. Math., 2 (1952), 25-53. | MR 54865 | Zbl 0046.10803

[11] Morrey, C. B. Jr., Multiple Integrals in the Calculus of Variations, Springer-Verlag, Berlin, 1966. | MR 202511 | Zbl 0142.38701

[12] Šverák, V., Rank-one-convexity does not imply quasiconvexity, Proc. Royal Soc. Edinburgh A, 120 (1992), 185-189. | MR 1149994 | Zbl 0777.49015