A-quasiconvexity : relaxation and homogenization
ESAIM: Control, Optimisation and Calculus of Variations, Tome 5 (2000), pp. 539-577.
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     title = {A-quasiconvexity : relaxation and homogenization},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {539--577},
     publisher = {EDP-Sciences},
     volume = {5},
     year = {2000},
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     url = {http://archive.numdam.org/item/COCV_2000__5__539_0/}
}
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Braides, Andrea; Fonseca, Irene; Leoni, Giovanni. A-quasiconvexity : relaxation and homogenization. ESAIM: Control, Optimisation and Calculus of Variations, Tome 5 (2000), pp. 539-577. http://archive.numdam.org/item/COCV_2000__5__539_0/

[1] E. Acerbi and N. Fusco, Semicontinuity problems in the calculus of variations. Arch. Rational Mech. Anal. 86 ( 1984) 125 -145. | MR | Zbl

[2] M. Amar and V. De Cicco, Relaxation of quasi-convex integrals of arbitrary order. Proc. Roy. Soc. Edinburgh Sect. A 124 ( 1994) 927-946. | MR | Zbl

[3] L. Ambrosio, S. Mortola and V.M. Tortorelli, Functionals with linear growth defined on vector valued BV functions. J. Math. Pures Appl. 70 ( 1991) 269-323. | MR | Zbl

[4] E.J. Balder, A general approach to lower semicontinuity and lower closure in optimal control theory. SIAM J. Control Optim. 22 ( 1984) 570-598. | MR | Zbl

[5] J.M. Ball, A version of the fundamental theorem for Young measures, in PDE's and Continuum Models of Phase Transitions, edited by M. Rascle, D. Serre and M. Slemrod. Springer-Verlag, Berlin, Lecture Notes in Phys. 344 ( 1989) 207-215. | MR | Zbl

[6] J.M. Ball and F. Murat, Remarks on Chacon's biting lemma. Proc. Amer. Math. Soc. 107 ( 1989) 655-663. | MR | Zbl

[7] H. Berliocchi and J.M. Lasry, Intégrands normales et mesures paramétrées en calcul des variations. Bull. Soc. Math. France 101 ( 1973) 129-184. | Numdam | MR | Zbl

[8] A. Braides, A homogenization theorem for weakly almost periodic functionals, Rend. Accad. Naz. Sci. XL Mem. Sci. Fis. Natur. (5) 104 ( 1986) 261-281. | MR | Zbl

[9] A. Braides, Relaxation of functionals with constraints on the divergence. Ann. Univ. Ferrara Ser. VII (N.S.) 33 ( 1987) 157-177. | MR | Zbl

[10] A. Braides and A. Defranceschi, Homogenization of Multiple Integrals. Clarendon Press, Oxford ( 1998). | MR | Zbl

[11] A. Braides, A. Defranceschi and E. Vitali, Homogenization of free discontinuity problems. Arch. Rational Mech. Anal. 135 ( 1996) 297-356. | MR | Zbl

[12] G. Buttazzo, Semicontinuity, relaxation and integral epresentation problems in the Calculus of Variations. Longman, Harlow, Pitman Res. Notes Math. Ser. 207 ( 1989). | Zbl

[13] B. Dacorogna, Direct Methods in the Calculus of Variations. Springer-Verlag, Berlin ( 1989). | MR | Zbl

[14] B. Dacorogna, Weak Continuity and Weak Lower Semicontinuity for Nonlinear Functionals. Springer-Verlag, Berlin, Lecture Notes in Math. 922 ( 1982). | MR | Zbl

[15] G. Dal Maso, An Introduction to Γ-Convergence. Birkhäuser, Boston ( 1993). | MR | Zbl

[16] G. Dal Maso, A. Defranceschi and E. Vitali (private communication).

[17] A. De Simone, Energy minimizers for large ferromagnetic bodies. Arch. Rational Mech. Anal. 125 ( 1993) 99-143. | MR | Zbl

[18] I. Fonseca, The lower quasiconvex envelope of the stored energy function for an elastic crystal. J. Math. Pures Appl. 67 ( 1988) 175-195. | MR | Zbl

[19] I. Fonseca, G. Leoni, J. Malý and R. Paroni (in preparation).

[20] I. Fonseca and S. Müller, Quasiconvex integrands and lower semicontinuity in L1. SIAM J. Math. Anal. 23 ( 1992) 1081-1098. | MR | Zbl

[21] I. Fonseca and S. Müller, Relaxation of quasiconvex functionals in BV(Ω, ℝp) for integrands f (x, u, ∆u). Arch. Rational Mech. Anal. 123 ( 1993) 1-49. | MR | Zbl

[22] I. Fonseca and S. Müller, A-quasiconvexity, lower semicontinuity and Young measures. SIAM J. Math. Anal. 30 ( 1999) 1355-1390. | MR | Zbl

[23] N. Fusco, Quasi-convessitá e semicontinuitá per integrali di ordine superiore. Ricerche Mat. 29 ( 1980) 307-323. | Zbl

[24] M. Giaquinta and G. Modica, Regularity results for some classes of higher order non linear elliptic systems. J. reine angew. Math. 311/312 ( 1979) 145-169. | MR | Zbl

[25] M. Guidorzi and L. Poggiolini, Lower semicontinuity for quasiconvex integrals of higher order. NoDEA Nonlinear Differential Equations Appl. 6 ( 1999) 227-246. | MR | Zbl

[26] J. Kristensen, Finite functionals and Young measures generated by gradients of Sobolev functions. Mathematical Institute, Technical University of Denmark, Mat-Report No. 1994-34 ( 1994).

[27] P. Marcellini, Approximation of quasiconvex functions and semicontinuity of multiple integrals. Manuscripta Math. 51 ( 1985) 1-28. | MR | Zbl

[28] P. Marcellini and C. Sbordone, Semicontinuity problems in the Calculus of Variations. Nonlinear Anal. 4 ( 1980) 241-257. | MR | Zbl

[29] N.G. Meyers, Quasi-convexity and lower semi-continuity of multiple variational integrals of any order. Trans. Amer. Math. Soc. 119 ( 1965) 125-149. | MR | Zbl

[30] C.B. Morrey, Multiple Integrals in the calculus of Variations. Springer-Verlag, Berlin ( 1966). | MR | Zbl

[31] S. Müller, Variational models for microstructures and phase transitions, in Calculus of Variations and Geometric Evolution Problems, edited by S. Hildebrant et al. Springer-Verlag, Berlin, Lecture Notes in Math. 1713 ( 1999) 85-210. | MR | Zbl

[32] F. Murat, Compacité par compensation : condition nécessaire et suffisante de continuité faible sous une hypothèse de rang constant. Ann. Scuola Norm. Sup. Pisa Cl. Sw. (4) 8 ( 1981) 68-102. | Numdam | MR | Zbl

[33] P. Pedregal, Parametrized Measures and Variational Principles. Birkhäuser, Boston ( 1997). | MR | Zbl

[34] L. Tartar, Compensated compactness and applications to partial differential equations, in Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, edited by R. Knops. Longman, Harlow, Pitman Res. Notes Math. Ser. 39 ( 1979) 136-212. | MR | Zbl

[35] L. Tartar, The compensated compactness method applied to systems of conservation laws, in Systems of Nonlinear Partial Differential Eq., edited by J.M. Ball. Riedel ( 1983). | MR | Zbl

[36] L. Tartar, Étude des oscillations dans les équations aux dérivées partielles nonlinéaires. Springer-Verlag, Berlin, Lectures Notes in Phys. 195 ( 1984) 384-412. | MR | Zbl

[37] L. Tartar, H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations. Proc. Roy. Soc. Edinburgh Sect. A 115 ( 1990) 193-230. | MR | Zbl

[38] L. Tartar, On mathematical tools for studying partial differential equations of continuum physics: H-measures and Young measures, in Developments in Partial Differential Equations and Applications to Mathematical Physics, edited by Buttazzo, Galdi and Zanghirati. Plenum, New York ( 1991). | MR | Zbl

[39] L. Tartar, Some remarks on separately convex functions, in Microstructure and Phase Transitions, edited by D. Kinderlehrer, R.D. James, M. Luskin and J.L. Ericksen. Springer-Verlag, IMA J. Math. Appl. 54 ( 1993) 191-204. | MR | Zbl

[40] L.C. Young, Lectures on Calculus of Variations and Optimal Control Theory. W.B. Saunders ( 1969). | MR | Zbl