Existence of minimizers for non-quasiconvex functionals arising in optimal design
Annales de l'I.H.P. Analyse non linéaire, Tome 15 (1998) no. 3, p. 301-339
@article{AIHPC_1998__15_3_301_0,
     author = {Allaire, Gr\'egoire and Francfort, Gilles},
     title = {Existence of minimizers for non-quasiconvex functionals arising in optimal design},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Gauthier-Villars},
     volume = {15},
     number = {3},
     year = {1998},
     pages = {301-339},
     zbl = {0913.49008},
     mrnumber = {1629349},
     language = {en},
     url = {http://http://www.numdam.org/item/AIHPC_1998__15_3_301_0}
}
Allaire, Grégoire; Francfort, Gilles. Existence of minimizers for non-quasiconvex functionals arising in optimal design. Annales de l'I.H.P. Analyse non linéaire, Tome 15 (1998) no. 3, pp. 301-339. http://www.numdam.org/item/AIHPC_1998__15_3_301_0/

[1] G. Allaire, E. Bonnetier, G.A. Francfort and F. Jouve, Shape optimization by the homogenization method, Num. Math., Vol. 76, 1997, pp. 27-68. | MR 1438681 | Zbl 0889.73051

[2] G. Allaire and R.K. Kohn, Optimal bounds on the effective behavior of a mixture of two well-ordered elastic materials, Quat. Appl. Math., Vol. 51, 1993, pp. 643-674. | MR 1247433 | Zbl 0805.73043

[3] M. Avellaneda, Optimal bounds and microgeometries for elastic two-phase composites, SIAM, J. Appl. Math., Vol. 47, 6, 1987, pp. 1216-1228. | MR 916238 | Zbl 0632.73079

[4] J.M. Ball and R.D. James, Fine phase mixtures as minimizers of energy, Arch. Rat. Mech. Anal., Vol. 100, 1, 1987, pp. 13-52. | MR 906132 | Zbl 0629.49020

[5] J.M. Ball and F. Murat, W1,p quasiconvexity and variational problems for multiple integrals, J. Func. Anal., Vol. 58, 1984, pp. 225-253. | MR 759098 | Zbl 0549.46019

[6] B. Dacorogna, Direct Methods in the Calculus of Variations, Springer Verlag, Berlin Heidelberg (1989). | MR 990890 | Zbl 0703.49001

[7] B. Dacorogna and P. Marcellini, Existence of minimizers for non quasiconvex integrals, Arch. Rational Mech. Anal., Vol. 131, 1995, pp. 359-399. | MR 1354700 | Zbl 0837.49002

[8] G. Dal Maso and R.V. Kohn, The local character of G-closure, to appear.

[9] I. Fonseca and G. Francfort, Relaxation in BV versus quasiconvexification in W1,p; a model for the interaction between fracture and damage, Calculus of Variations, Vol. 3, 4, 1995, pp. 407-446. | MR 1385294 | Zbl 0847.73077

[10] I. Fonseca and S. Müller, A-quasiconvexity: a necessary and sufficient condition for Lp weak lower semicontinuity under p.d.e. constraints, to appear.

[11] R.V. Kohn and G. Strang, Optimal design and relaxation of variational problem I, II, III, Comm. Pure and Appl. Math., Vol. 39, 1986, pp. 353-377. | MR 829845 | Zbl 0694.49004

[12] P. Marcellini, Periodic solutions and homogenization of nonlinear variational problems, Ann. Mat. Pura Appl., Vol. 1178, 1978, pp. 139-152. | MR 515958 | Zbl 0395.49007

[13] L. Mirsky, On the trace of a matrix product, Math. Nachr. Vol. 20, 1959, pp. 171-174. | MR 125851 | Zbl 0136.24901

[14] F. Murat, Contre-exemples pour divers problèmes où le contrôle intervient dans les coefficients, Ann. Mat. Pura Appl., Vol. 112, 1977, pp. 49-68. | MR 438205 | Zbl 0349.49005

[15] F. Murat and L. Tartar, H-convergence, to appear in Topics in the mathematical modeling of composite materials, R. V. Koh, ed., series: Progress in Nonlinear Differential Equations and their Applications, Birkhaüser, Boston (french version: mimeographed notes, séminaire d'Analyse Fonctionnelle et Numérique de l'Université d'Alger, 1978. | MR 1493039 | Zbl 0920.35019

[16] F. Murat and L. Tartar, Calcul des variations et Homogénéisation, Les Méthodes de l'Homogénéisation Théorie et Applications en Physique, Coll. Dir. Études et Recherches EDF, Eyrolles, 1985, pp. 319-369. | MR 844873

[17] L. Tartar, Estimations fines de coefficients homogénéisés, Ennio de Giorgi Colloquium, P. Krée ed., Pitman Research Notes in Math., Vol. 125, 1985, pp. 168-187. | MR 909716 | Zbl 0586.35004