Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-06T01:08:47.459Z Has data issue: false hasContentIssue false
  • English
  • Français

Homogenization of periodic nonconvex integral functionals in terms of Young measures

Published online by Cambridge University Press:  15 December 2005

Omar Anza Hafsa ,
Jean-Philippe Mandallena  and
Gérard Michaille
Omar Anza Hafsa
Affiliation:
Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland; [email protected]
Jean-Philippe Mandallena
Affiliation:
EMIAN (Équipe de Mathématiques, d'Informatiques et Applications de Nîmes), Centre Universitaire de Formation et de Recherche de Nîmes, Site des Carmes, Place Gabriel Péri, Cedex 01, 30021 Nîmes, France; [email protected] I3M (Institut de Mathématiques et Modélisation de Montpellier) UMR-CNRS 5149, Université Montpellier II, Place Eugène Bataillon, 34090 Montpellier, France; [email protected]
Gérard Michaille
Affiliation:
EMIAN (Équipe de Mathématiques, d'Informatiques et Applications de Nîmes), Centre Universitaire de Formation et de Recherche de Nîmes, Site des Carmes, Place Gabriel Péri, Cedex 01, 30021 Nîmes, France; [email protected] I3M (Institut de Mathématiques et Modélisation de Montpellier) UMR-CNRS 5149, Université Montpellier II, Place Eugène Bataillon, 34090 Montpellier, France; [email protected]
Get access

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.

Keywords

Young measureshomogenization.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 12 , Issue 1 , January 2006 , pp. 35 - 51
Copyright
© EDP Sciences, SMAI, 2006

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Akcoglu, M.A. and Krengel, U., Ergodic theorems for superadditive processes. J. Reine Angew. Math. 323 (1981) 53–67.
Alvarez, F. and Mandallena, J.-P., Homogenization of multiparameter integrals. Nonlinear Anal. 50 (2002) 839–870. CrossRef
H. Attouch, Variational convergence for functions and operators. Pitman (1984).
Ball, J.M. and James, R.D., Fine phase mixtures as minimizers of energy. Arch. Rat. Mech. Anal. 100 (1987) 13–52. CrossRef
Bhattacharya, K. and Kohn, R., Elastic energy minimization and the recoverable strains of polycristalline shape-memory materials. Arch. Rat. Mech. Anal. 139 (1997) 99–180. CrossRef
Braides, A., Homogenization of some almost periodic coercive functional. Rend. Accad. Naz. Sci. 103 (1985) 313–322.
A. Braides and A. Defranceschi, Homogenization of multiple integrals. Oxford University Press (1998).
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).
C. Castaing and M. Valadier, Convex analysis and measurable multifunctions. Lect. Notes Math. 580 (1977).
Dacorogna, B., Quasiconvexity and relaxation of nonconvex variational problems. J. Funct. Anal. 46 (1982) 102–118. CrossRef
G. Dal maso, An introduction to Γ-convergence. Birkhäuser (1993).
Dal, G. maso and L. Modica, Nonlinear stochastic homogenization. J. Reine Angew. Math. 363 (1986) 27–43.
L.C. Evans, Weak convergence methods for nonlinear partial differential equations. CBMS Amer. Math. Soc. 74 (1990).
Fonseca, I., Müller, S. and Pedregal, P., Analysis of concentration and oscillation effects generated by gradients. SIAM J. Math. Anal. 29 (1998) 736–756. CrossRef
Kinderlherer, D. and Pedregal, P., Characterization of Young measure generated by gradients. Arch. Rat. Mech. Anal. 115 (1991) 329–365. CrossRef
Kinderlherer, D. and Pedregal, P., Gradient Young measures generated by sequences in Sobolev spaces. J. Geom. Anal. 4 (1994) 59–89. CrossRef
Licht, C. and Michaille, G., Global-local subadditive ergodic theorems and application to homogenization in elasticity. Ann. Math. Blaise Pascal 9 (2002) 21–62. CrossRef
Marcellini, P., Periodic solutions and homogenization of nonlinear variational problems. Annali Mat. Pura Appl. 117 (1978) 139–152. CrossRef
Müller, S., Homogenization of nonconvex integral functionals and cellular elastic materials. Arch. Rat. Mech. Anal. 100 (1987) 189–212.
P. Pedregal, Parametrized measures and variational principles. Birkhäuser (1997).
Pedregal, P., Γ-convergence through Young meaasures. SIAM J. Math. Anal. 36 (2004) 423–440. CrossRef
Valadier, M., Young measures. Lect. Notes Math. 1446 (1990) 152–188. CrossRef
M. Valadier, A course on Young measures. Rend. Istit. Mat. Univ. Trieste 26 (1994) Suppl. 349–394.
W.P. Ziemer, Weakly differentiable functions. Springer (1989).