A-quasiconvexity : relaxation and homogenization

Andrea Braides; Irene Fonseca; Giovanni Leoni

ESAIM: Control, Optimisation and Calculus of Variations (2000)

  • Volume: 5, page 539-577
  • ISSN: 1292-8119

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Braides, Andrea, Fonseca, Irene, and Leoni, Giovanni. "A-quasiconvexity : relaxation and homogenization." ESAIM: Control, Optimisation and Calculus of Variations 5 (2000): 539-577. <http://eudml.org/doc/90582>.

@article{Braides2000,
author = {Braides, Andrea, Fonseca, Irene, Leoni, Giovanni},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {equi-integrability; Young measures; Gamma-convergence; differential constraint; integral representation of the relaxed energy},
language = {eng},
pages = {539-577},
publisher = {EDP Sciences},
title = {A-quasiconvexity : relaxation and homogenization},
url = {http://eudml.org/doc/90582},
volume = {5},
year = {2000},
}

TY - JOUR
AU - Braides, Andrea
AU - Fonseca, Irene
AU - Leoni, Giovanni
TI - A-quasiconvexity : relaxation and homogenization
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2000
PB - EDP Sciences
VL - 5
SP - 539
EP - 577
LA - eng
KW - equi-integrability; Young measures; Gamma-convergence; differential constraint; integral representation of the relaxed energy
UR - http://eudml.org/doc/90582
ER -

References

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  1. [1] E. Acerbi and N. Fusco, Semicontinuity problems in the calculus of variations. Arch. Rational Mech. Anal. 86 ( 1984) 125 -145. Zbl0565.49010MR751305
  2. [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. Zbl0831.49025MR1303762
  3. [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. Zbl0662.49007MR1113814
  4. [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. Zbl0549.49005MR747970
  5. [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. Zbl0991.49500MR1036070
  6. [6] J.M. Ball and F. Murat, Remarks on Chacon's biting lemma. Proc. Amer. Math. Soc. 107 ( 1989) 655-663. Zbl0678.46023MR984807
  7. [7] H. Berliocchi and J.M. Lasry, Intégrands normales et mesures paramétrées en calcul des variations. Bull. Soc. Math. France 101 ( 1973) 129-184. Zbl0282.49041MR344980
  8. [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. Zbl0611.49007MR879115
  9. [9] A. Braides, Relaxation of functionals with constraints on the divergence. Ann. Univ. Ferrara Ser. VII (N.S.) 33 ( 1987) 157-177. Zbl0662.49004MR958391
  10. [10] A. Braides and A. Defranceschi, Homogenization of Multiple Integrals. Clarendon Press, Oxford ( 1998). Zbl0911.49010MR1684713
  11. [11] A. Braides, A. Defranceschi and E. Vitali, Homogenization of free discontinuity problems. Arch. Rational Mech. Anal. 135 ( 1996) 297-356. Zbl0924.35015MR1423000
  12. [12] G. Buttazzo, Semicontinuity, relaxation and integral epresentation problems in the Calculus of Variations. Longman, Harlow, Pitman Res. Notes Math. Ser. 207 ( 1989). Zbl0669.49005
  13. [13] B. Dacorogna, Direct Methods in the Calculus of Variations. Springer-Verlag, Berlin ( 1989). Zbl0703.49001MR990890
  14. [14] B. Dacorogna, Weak Continuity and Weak Lower Semicontinuity for Nonlinear Functionals. Springer-Verlag, Berlin, Lecture Notes in Math. 922 ( 1982). Zbl0484.46041MR658130
  15. [15] G. Dal Maso, An Introduction to Γ-Convergence. Birkhäuser, Boston ( 1993). Zbl0816.49001MR1201152
  16. [16] G. Dal Maso, A. Defranceschi and E. Vitali (private communication). 
  17. [17] A. De Simone, Energy minimizers for large ferromagnetic bodies. Arch. Rational Mech. Anal. 125 ( 1993) 99-143. Zbl0811.49030MR1245068
  18. [18] I. Fonseca, The lower quasiconvex envelope of the stored energy function for an elastic crystal. J. Math. Pures Appl. 67 ( 1988) 175-195. Zbl0718.73075MR949107
  19. [19] I. Fonseca, G. Leoni, J. Malý and R. Paroni (in preparation). 
  20. [20] I. Fonseca and S. Müller, Quasiconvex integrands and lower semicontinuity in L1. SIAM J. Math. Anal. 23 ( 1992) 1081-1098. Zbl0764.49012MR1177778
  21. [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. Zbl0788.49039MR1218685
  22. [22] I. Fonseca and S. Müller, A-quasiconvexity, lower semicontinuity and Young measures. SIAM J. Math. Anal. 30 ( 1999) 1355-1390. Zbl0940.49014MR1718306
  23. [23] N. Fusco, Quasi-convessitá e semicontinuitá per integrali di ordine superiore. Ricerche Mat. 29 ( 1980) 307-323. Zbl0508.49012
  24. [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. Zbl0409.35015MR549962
  25. [25] M. Guidorzi and L. Poggiolini, Lower semicontinuity for quasiconvex integrals of higher order. NoDEA Nonlinear Differential Equations Appl. 6 ( 1999) 227-246. Zbl0930.35059MR1691445
  26. [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. [27] P. Marcellini, Approximation of quasiconvex functions and semicontinuity of multiple integrals. Manuscripta Math. 51 ( 1985) 1-28. Zbl0573.49010MR788671
  28. [28] P. Marcellini and C. Sbordone, Semicontinuity problems in the Calculus of Variations. Nonlinear Anal. 4 ( 1980) 241-257. Zbl0537.49002MR563807
  29. [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. Zbl0166.38501MR188838
  30. [30] C.B. Morrey, Multiple Integrals in the calculus of Variations. Springer-Verlag, Berlin ( 1966). Zbl0142.38701MR202511
  31. [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. Zbl0968.74050MR1731640
  32. [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. Zbl0464.46034MR616901
  33. [33] P. Pedregal, Parametrized Measures and Variational Principles. Birkhäuser, Boston ( 1997). Zbl0879.49017MR1452107
  34. [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. Zbl0437.35004MR584398
  35. [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). Zbl0536.35003MR725524
  36. [36] L. Tartar, Étude des oscillations dans les équations aux dérivées partielles nonlinéaires. Springer-Verlag, Berlin, Lectures Notes in Phys. 195 ( 1984) 384-412. Zbl0595.35012MR755737
  37. [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. Zbl0774.35008MR1069518
  38. [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). Zbl0897.35010MR1213932
  39. [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. Zbl0823.26008MR1320538
  40. [40] L.C. Young, Lectures on Calculus of Variations and Optimal Control Theory. W.B. Saunders ( 1969). Zbl0177.37801MR259704

Citations in EuDML Documents

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  1. Giovanni Pisante, Homogenization of micromagnetics large bodies
  2. Giovanni Pisante, Homogenization of micromagnetics large bodies
  3. Nadia Ansini, Adriana Garroni, -convergence of functionals on divergence-free fields
  4. Stefan Krömer, Dimension reduction for functionals on solenoidal vector fields
  5. Stefan Krömer, Dimension reduction for functionals on solenoidal vector fields
  6. Irene Fonseca, Giovanni Leoni, Stefan Müller, A-quasiconvexity : weak-star convergence and the gap
  7. Irene Fonseca, Martin Kružík, Oscillations and concentrations generated by 𝒜 -free mappings and weak lower semicontinuity of integral functionals

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