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. Giuliano Gargiulo, Elvira Zappale, The Energy Density of Non Simple Materials Grade Two Thin Films via a Young Measure Approach
  6. Stefan Krömer, Dimension reduction for functionals on solenoidal vector fields
  7. Irene Fonseca, Giovanni Leoni, Stefan Müller, A-quasiconvexity : weak-star convergence and the gap
  8. Irene Fonseca, Martin Kružík, Oscillations and concentrations generated by 𝒜 -free mappings and weak lower semicontinuity of integral functionals

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