Licht, Christian; Michaille, Gérard Global-local subadditive ergodic theorems and application to homogenization in elasticity. (English) Zbl 1070.28006 Ann. Math. Blaise Pascal 9, No. 1, 21-62 (2002). A global-local ergodic theorem about subadditive processes is established and applied in order to identify some limit problems in homogenization theory in presence of several small parameters. The invariant case is first examined. Then, the global result is extended to the quasiperiodic case. The random case is also treated, and the Ackoglu-Krengel ergodic theorem is recovered. When the subadditive process is parametrized in a separable space, it is shown that the convergence takes place in the variational sense of \(\Gamma\)-convergence. Applications are given in the setting of nonlinear elasticity. Reviewer: Riccardo De Arcangelis (Napoli) Cited in 21 Documents MSC: 28D05 Measure-preserving transformations 37A30 Ergodic theorems, spectral theory, Markov operators 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure 47A35 Ergodic theory of linear operators 74Q05 Homogenization in equilibrium problems of solid mechanics 49J45 Methods involving semicontinuity and convergence; relaxation 74B20 Nonlinear elasticity Keywords:ergodic theory; homogenization; subadditive processes; nonlinear elasticity × Cite Format Result Cite Review PDF Full Text: DOI Numdam EuDML References: [1] Abddaimi, Y.. Homogénéisation de quelques problèmes en analyse variationnelle, application des théorèmes ergodiques sous-additifs. Thèse, Univerité Montpellier 2, 1996. [2] Abddaimi, Y., Licht, C., and Michaille, G.. Stochastic homogenization for an integral functional of quasiconvex function with linear growth. Asymptotic Analysis, 15:183-202, 1997. · Zbl 0912.49013 [3] Ackoglu, M.A. and Krengel, U.. Ergodic theorems for superadditive processes. J. Reine angew. Math., 323:53-67, 1981. · Zbl 0453.60039 [4] Attouch, H.. Variational Convergence for Functions and Operators. Pitman Advanced Publishing Program, London, 1985. · Zbl 0561.49012 [5] Attouch, H. and Wets, R J.B.. Epigraphical processes : laws of large numbers for random lsc functions. Séminaire d’Analyse Convexe, 13, 1990. · Zbl 0744.60021 [6] Bellieud, M. and Bouchitté, G.. Homogenization of elliptic problems in a fiber reinforced structure. Ann. Scuola Norm. Sup. Pisa, Serie IV, XXVI, 4, 1998. · Zbl 0919.35014 [7] Bouchitté, G., Fonseca, I., and Mascarenhas, L.. A global method for relaxation. Arch. Rational Mech. Anal., 145:51-98, 1998. · Zbl 0921.49004 [8] Braides, A.. Homogenization of some almost-periodic functional. Rend. Accad. Naz. XL, 103:313-322, 1985. · Zbl 0582.49014 [9] Dacorogna, B.. Direct methods in the Calculus of Variations. Springer-Verlag, Berlin, 1989. · Zbl 0703.49001 [10] Hess, C.. Epi-convergence of sequences of normal integrands and strong consistency of the maximum likelihood estimator. The Annals of Statistics, 24:1298-1315, 1996. · Zbl 0862.62029 [11] Krengel, U.. Ergodic Theorems. Walter de Gruyter, Berlin, New York, 1985. · Zbl 0575.28009 [12] Licht, C. and Michaille, G.. Une modélisation du comportement d’un joint collé élastique. C.R. Acad. Sci. Paris, 322, Série I:295-300, 1996. · Zbl 0863.73019 [13] Licht, C. and Michaille, G.. A modelling of elastic adhesive bonding joints. Mathematical Sciences and Applications, 7:711-740, 1997. · Zbl 0892.73007 [14] Dal Maso, G.and Modica, L.. Non linear stochastic homogenization and ergodic theory. J. Reine angew. Math., 363:27-43, 1986. · Zbl 0582.60034 [15] Michaille, G., Michel, J., and Piccinini, L.. Large deviations estimates for epigraphical superadditive processes in stochastic homogenization. prepublication ENSLyon, 220, 1998. [16] Muller, S.. Homogenization of non convex integral functionals and cellular elastic material. Arch. Rational Mech. Anal., 7:189-212, 1987. · Zbl 0629.73009 [17] Rockafellar, R.T.. Integral functionals, normal integrands and measurable selections. , 543:133-158, 1979. · Zbl 0374.49001 [18] Nguyen Xuhan XanhZessin, H.. Ergodic theorems for spatial processes. Z. Wah. Verw. Gebiete, 48:133-158, 1979. · Zbl 0397.60080 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.