Homogenization of periodic nonconvex integral functionals in terms of Young measures
Omar Anza Hafsa; Jean-Philippe Mandallena; Gérard Michaille
ESAIM: Control, Optimisation and Calculus of Variations (2006)
- Volume: 12, Issue: 1, page 35-51
- ISSN: 1292-8119
Access Full Article
topAbstract
topHow to cite
topHafsa, Omar Anza, Mandallena, Jean-Philippe, and Michaille, Gérard. "Homogenization of periodic nonconvex integral functionals in terms of Young measures." ESAIM: Control, Optimisation and Calculus of Variations 12.1 (2006): 35-51. <http://eudml.org/doc/244977>.
@article{Hafsa2006,
abstract = {Homogenization of periodic functionals, whose integrands possess possibly multi-well structure, is treated in terms of Young measures. More precisely, we characterize the $\Gamma $-limit of sequences of such functionals in the set of Young measures, extending the relaxation theorem of Kinderlherer and Pedregal. We also make precise the relationship between our homogenized density and the classical one.},
author = {Hafsa, Omar Anza, Mandallena, Jean-Philippe, Michaille, Gérard},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Young measures; homogenization},
language = {eng},
number = {1},
pages = {35-51},
publisher = {EDP-Sciences},
title = {Homogenization of periodic nonconvex integral functionals in terms of Young measures},
url = {http://eudml.org/doc/244977},
volume = {12},
year = {2006},
}
TY - JOUR
AU - Hafsa, Omar Anza
AU - Mandallena, Jean-Philippe
AU - Michaille, Gérard
TI - Homogenization of periodic nonconvex integral functionals in terms of Young measures
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2006
PB - EDP-Sciences
VL - 12
IS - 1
SP - 35
EP - 51
AB - Homogenization of periodic functionals, whose integrands possess possibly multi-well structure, is treated in terms of Young measures. More precisely, we characterize the $\Gamma $-limit of sequences of such functionals in the set of Young measures, extending the relaxation theorem of Kinderlherer and Pedregal. We also make precise the relationship between our homogenized density and the classical one.
LA - eng
KW - Young measures; homogenization
UR - http://eudml.org/doc/244977
ER -
References
top- [1] M.A. Akcoglu and U. Krengel, Ergodic theorems for superadditive processes. J. Reine Angew. Math. 323 (1981) 53–67. Zbl0453.60039
- [2] F. Alvarez and J.-P. Mandallena, Homogenization of multiparameter integrals. Nonlinear Anal. 50 (2002) 839–870. Zbl1005.49008
- [3] H. Attouch, Variational convergence for functions and operators. Pitman (1984). Zbl0561.49012MR773850
- [4] J.M. Ball and R.D. James, Fine phase mixtures as minimizers of energy. Arch. Rat. Mech. Anal. 100 (1987) 13–52. Zbl0629.49020
- [5] K. Bhattacharya and R. Kohn, Elastic energy minimization and the recoverable strains of polycristalline shape-memory materials. Arch. Rat. Mech. Anal. 139 (1997) 99–180. Zbl0894.73225
- [6] A. Braides, Homogenization of some almost periodic coercive functional. Rend. Accad. Naz. Sci. 103 (1985) 313–322. Zbl0582.49014
- [7] A. Braides and A. Defranceschi, Homogenization of multiple integrals. Oxford University Press (1998). Zbl0911.49010MR1684713
- [8] C. Castaing, P. Raynaud de Fitte and M. Valadier, Young measures on topological spaces with applications in control theory and probability theory. Mathematics and Its Applications, Kluwer, The Netherlands (2004). Zbl1067.28001MR2102261
- [9] C. Castaing and M. Valadier, Convex analysis and measurable multifunctions. Lect. Notes Math. 580 (1977). Zbl0346.46038MR467310
- [10] B. Dacorogna, Quasiconvexity and relaxation of nonconvex variational problems. J. Funct. Anal. 46 (1982) 102–118. Zbl0547.49003
- [11] G. Dalmaso, An introduction to -convergence. Birkhäuser (1993). Zbl0816.49001MR1201152
- [12] G. Dal maso and L. Modica, Nonlinear stochastic homogenization. J. Reine Angew. Math. 363 (1986) 27–43. Zbl0607.49010
- [13] L.C. Evans, Weak convergence methods for nonlinear partial differential equations. CBMS Amer. Math. Soc. 74 (1990). Zbl0698.35004MR1034481
- [14] I. Fonseca, S. Müller and P. Pedregal, Analysis of concentration and oscillation effects generated by gradients. SIAM J. Math. Anal. 29 (1998) 736–756. Zbl0920.49009
- [15] D. Kinderlherer and P. Pedregal, Characterization of Young measure generated by gradients. Arch. Rat. Mech. Anal. 115 (1991) 329–365. Zbl0754.49020
- [16] D. Kinderlherer and P. Pedregal, Gradient Young measures generated by sequences in Sobolev spaces. J. Geom. Anal. 4 (1994) 59–89. Zbl0808.46046
- [17] C. Licht and G. Michaille, Global-local subadditive ergodic theorems and application to homogenization in elasticity. Ann. Math. Blaise Pascal 9 (2002) 21–62. Zbl1070.28006
- [18] P. Marcellini, Periodic solutions and homogenization of nonlinear variational problems. Annali Mat. Pura Appl. 117 (1978) 139–152. Zbl0395.49007
- [19] S. Müller, Homogenization of nonconvex integral functionals and cellular elastic materials. Arch. Rat. Mech. Anal. 100 (1987) 189–212. Zbl0629.73009
- [20] P. Pedregal, Parametrized measures and variational principles. Birkhäuser (1997). Zbl0879.49017MR1452107
- [21] P. Pedregal, -convergence through Young meaasures. SIAM J. Math. Anal. 36 (2004) 423–440. Zbl1077.49012
- [22] M. Valadier, Young measures. Lect. Notes Math. 1446 (1990) 152–188. Zbl0738.28004
- [23] M. Valadier, A course on Young measures. Rend. Istit. Mat. Univ. Trieste 26 (1994) Suppl. 349–394. Zbl0880.49013
- [24] W.P. Ziemer, Weakly differentiable functions. Springer (1989). Zbl0692.46022MR1014685
Citations in EuDML Documents
topNotesEmbed ?
topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.