×

Nonlinear stochastic homogenization. (English) Zbl 0607.49010

In the framework of calculus of variations, the authors study stochastic homogenization of integral functionals. The integrand is supposed measurable in the first variable and convex in the second one. The main feature of this paper is to pass from the view-point of the stochastic differential equations to be solved to that one of the random integral functionals to be minimized. In order to study the random integral functionals, that are ”measurable” maps \(\omega\to F(\omega)\) from a probabilistic space \(\Omega\) into the functional class \({\mathcal F}\), and to study their convergence, a topological structure is given on \({\mathcal F}.\)
The applications of this result are concerned with a large number of real phenomena in physics, chemistry and engineering, where the structures to be homogenized are not periodic but only stochastically periodic.
Reviewer: M.Codegone

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
49J55 Existence of optimal solutions to problems involving randomness
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
49K45 Optimality conditions for problems involving randomness
Full Text: DOI

References:

[1] Bensoussan, A.; Lions, J. L.; Papanicolaou, G. C., Asymptotic Methods in Periodic Structures (1978), Amsterdam: North-Holland, Amsterdam · Zbl 0411.60078
[2] Brézis, H., Opérateurs Maximaux Monotones et Semigroupes de Contractions dans les Espaces de Hilbert, Notas de Matemàtica (50), Mathematics Studies (1973), Amsterdam: North-Holland, Amsterdam · Zbl 0252.47055
[3] Burridge, R., Microscopic Properties of Disordered Media (1981), Berlin-Heidelberg-New York: Springer-Verlag, Berlin-Heidelberg-New York
[4] Buttazzo, G.; Dal Maso, G., Γ-limits of Integral Functional, J. Analyse Math., 37, 145-185 (1980) · Zbl 0446.49012
[5] Dal Maso, G.; Modica, L.; Strocchi, F., A General Theory of Variational Functionals, Topics in Functional Analysis 1980-81, 149-221 (1981), Pisa: Scuola Normale Superiore, Pisa · Zbl 0493.49005
[6] De Giorgi, E.; Strocchi, F., Generalized Limits in Calculus of Variations, Topics in Functional Analysis 1980-81, 117-148 (1981), Pisa: Ann. Scuola Norm. Sup., Pisa · Zbl 0493.49004
[7] De Giorgi, E.; Franzoni, T., Su un tipo di convergenza variazionale, Atti Accad. Naz. Lincei, Rend. Cl. Sci. Mat. Fis. Natur., (8), 58, 842-850 (1975) · Zbl 0339.49005
[8] De Giorgi, E.; Spagnolo, S., Sulla convergenza degli integrali dell’energia per operatori ellittici del II ordine, Boll. Un. Mat. Ital., (4), 8, 391-411 (1973) · Zbl 0274.35002
[9] Ekeland, I.; Temam, R., Convex Analysis and Variational Problems, Studies in Mathematics and Its Applications, Vol. 1 (1976), Amsterdam: North-Holland, Amsterdam · Zbl 0322.90046
[10] G.Facchinetti,Dualità e convergenza stocastica dei minimi (to appear on Le Matematiche).
[11] Facchinetti, G.; Russo, L., Un caso unidimensionale di omogeneizzazione stocastica, Boll. Un. Mat. Ital., (6), 2-C, 159-170 (1983) · Zbl 0534.49011
[12] Federer, H., Geometric Measure Theory (1969), Berlin-Heidelberg-New York: Springer-Verlag, Berlin-Heidelberg-New York · Zbl 0176.00801
[13] Kozlov, S. M., Averaging of Random Operators, Math. U.S.S.R. Sbornik, 37, 167-180 (1980) · Zbl 0444.60047
[14] Kuratowski, K., Topology (1966), New York: Academic Press, New York · Zbl 0158.40802
[15] Loéve, M., Probability Theory (1955), Toronto: D. Van Nostrand, Toronto · Zbl 0066.10903
[16] Marcellini, P., Periodic Solutions and Homogenization of Nonlinear Variational Problems, Ann. Mat. Pura Appl., (4), 117, 139-152 (1978) · Zbl 0395.49007
[17] L.Modica,Omogeneizzazione con coefficient casuali, Atti Convegno su «Studio di Problemi-Limite dell’Analisi Funzionale», Bressanone, 1981, Pitagora, Bologna, (1982), pp. 155-165.
[18] F.Murat - L.Tartar, Private Communication.
[19] Papanicolaou, G. C.; Varadhan, S. R. S.; Fritz, J.; Lebowitz, J. L.; Szász, D., Boundary Value Problems with Rapidly Oscillating Random Coefficients, Colloquia Mathematica Societ. Janos Bolyai, 835-873 (1979), Amsterdam: North-Holland, Amsterdam · Zbl 0499.60059
[20] Rudin, W., Functional Analysis (1973), New York: McGraw-Hill, New York · Zbl 0253.46001
[21] Spagnolo, S., Sulla convergenza delle soluzioni di equazioni paraboliche ed ellittiche, Ann. Scuola Norm. Sup. Pisa, Cl. Sci., (3), 21, 657-699 (1967) · Zbl 0153.42103
[22] Treves, F., Basic Linear Partial Differential Equations (1975), New York: Academic Press, New York · Zbl 0305.35001
[23] Vainberg, M. M., Variational Methods for the Study of Nonlinear Operators (1964), San Francisco: Holden Day, San Francisco · Zbl 0122.35501
[24] Yurinskij, V. V., Averaging an Elliptic Boundary-Value Problem with Random Cofficients, Siberian Math. Journal, 21, 470-482 (1980) · Zbl 0453.35090
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.