# 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 (2005)

- Volume: 12, Issue: 1, page 35-51
- ISSN: 1292-8119

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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 (2005): 35-51. <http://eudml.org/doc/90789>.

@article{Hafsa2005,

abstract = {
Homogenization of periodic functionals, whose integrands possess possibly multi-well structure, is treated in terms of Young measures. More precisely, we characterize the Γ-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.; homogenization},

language = {eng},

month = {12},

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/90789},

volume = {12},

year = {2005},

}

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

DA - 2005/12//

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 Γ-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.; homogenization

UR - http://eudml.org/doc/90789

ER -

## References

top- M.A. Akcoglu and U. Krengel, Ergodic theorems for superadditive processes. J. Reine Angew. Math.323 (1981) 53–67. Zbl0453.60039
- F. Alvarez and J.-P. Mandallena, Homogenization of multiparameter integrals. Nonlinear Anal.50 (2002) 839–870. Zbl1005.49008
- H. Attouch, Variational convergence for functions and operators. Pitman (1984). Zbl0561.49012
- J.M. Ball and R.D. James, Fine phase mixtures as minimizers of energy. Arch. Rat. Mech. Anal.100 (1987) 13–52. Zbl0629.49020
- 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
- A. Braides, Homogenization of some almost periodic coercive functional. Rend. Accad. Naz. Sci.103 (1985) 313–322. Zbl0582.49014
- A. Braides and A. Defranceschi, Homogenization of multiple integrals. Oxford University Press (1998). Zbl0911.49010
- 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). Zbl1067.28001
- C. Castaing and M. Valadier, Convex analysis and measurable multifunctions. Lect. Notes Math.580 (1977). Zbl0346.46038
- B. Dacorogna, Quasiconvexity and relaxation of nonconvex variational problems. J. Funct. Anal.46 (1982) 102–118. Zbl0547.49003
- G. Dal maso, An introduction to Γ-convergence. Birkhäuser (1993). Zbl0816.49001
- G. Dal maso and L. Modica, Nonlinear stochastic homogenization. J. Reine Angew. Math.363 (1986) 27–43. Zbl0607.49010
- L.C. Evans, Weak convergence methods for nonlinear partial differential equations. CBMS Amer. Math. Soc.74 (1990).
- 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
- D. Kinderlherer and P. Pedregal, Characterization of Young measure generated by gradients. Arch. Rat. Mech. Anal.115 (1991) 329–365. Zbl0754.49020
- D. Kinderlherer and P. Pedregal, Gradient Young measures generated by sequences in Sobolev spaces. J. Geom. Anal.4 (1994) 59–89. Zbl0808.46046
- C. Licht and G. Michaille, Global-local subadditive ergodic theorems and application to homogenization in elasticity. Ann. Math. Blaise Pascal9 (2002) 21–62. Zbl1070.28006
- P. Marcellini, Periodic solutions and homogenization of nonlinear variational problems. Annali Mat. Pura Appl.117 (1978) 139–152. Zbl0395.49007
- S. Müller, Homogenization of nonconvex integral functionals and cellular elastic materials. Arch. Rat. Mech. Anal.100 (1987) 189–212. Zbl0629.73009
- P. Pedregal, Parametrized measures and variational principles. Birkhäuser (1997).
- P. Pedregal, Γ-convergence through Young meaasures. SIAM J. Math. Anal.36 (2004) 423–440.
- M. Valadier, Young measures. Lect. Notes Math.1446 (1990) 152–188. Zbl0738.28004
- M. Valadier, A course on Young measures. Rend. Istit. Mat. Univ. Trieste26 (1994) Suppl. 349–394. Zbl0880.49013
- W.P. Ziemer, Weakly differentiable functions. Springer (1989). Zbl0692.46022

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