Asymptotic behaviour and correctors for Dirichlet problems in perforated domains with homogeneous monotone operators

Dal Maso, Gianni; Murat, François

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 4, Tome 24 (1997) no. 2, pp. 239-290.

[1] H. Attouch, Variational Convergence for Functions and Operators., Pitman, London, 1984. | Zbl

[2] H. Attouch - C. Picard, Variational inequalities with varying obstacles: the general form of the limit problem, J. Funct. Anal. 50 (1983), 329-386.

[3] T. Bagby, Quasi topologies and rational approximation, J. Funct. Anal. 10 (1972), 259-268. | Zbl

[4] L. Boccardo - F. Murat, Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlinear Anal. 19 (1992), 581-597. | Zbl

[5] G. Buttazzo - G. Dal Maso, Shape optimizationforDirichletproblems: relaxed solutions and optimality conditions, Appl. Math. Optim. 23 (1991), 17-49. | Zbl

[6] G. Buttazzo - G. Dal Maso - U. Mosco, A derivation theorem for capacities with respect to a Radon measure, J. Funct. Anal. 71 (1987), 263-278. | Zbl

[7] Buttazzo G. - Dal Maso G. - U. Mosco, Asymptotic behaviour for Dirichlet problems in domains bounded by thin layers, Partial Differential Equations and the Calculus of Variations, Essays in Honor of Ennio De Giorgi, 193-249, Birkhäuser, Boston, 1989. | Zbl

[8] J. Casado Diaz, Homogenization of a quasi-linear problem with quadratic growth in perforated domains: an example, Ann. Inst. H. Poincaré Anal. Non Linéaire, to appear. | Numdam | Zbl

[9] J. Casado Diaz, Homogenization of general quasi-linear problems with quadratic growth in perforated domains, J. Math. Pures Appl., to appear. | Zbl

[10] J. Casado Diaz, Homogenization of Dirichlet problems for monotone operators in varying domains, Proc. Roy. Soc. Edinburgh Sect. A, to appear. | Zbl

[11] J. Casado Diaz, Existence of a sequence satisfying Cioranescu-Murat conditions in homogenization of Dirichlet problems in perforated domains, Rend. Mat., to appear. | Zbl

[12] J. Casado Diaz - A. Garroni, Asymptotic behaviour of nonlinear elliptic systems on varying domains, to appear. | Zbl

[13] M. Chipot - G. Dal Maso, Relaxed shape optimization: the case of nonnegative data for the Dirichlet problem, Adv. Math. Sci. Appl. 1 (1992), 47-81. | Zbl

[14] D. Cioranescu - F. Murat, Un terme étrange venu d'ailleurs, I and II. Nonlinear Partial Differential Equations and Their Applications, Collège de France Seminar. Vol. II, 98-138, and Vol. III, 154-178, Res. Notes in Math., 60 and 70, Pitman, London, 1982 and 1983. English translation: A strange term coming from nowhere. Topics in the Mathematical Modelling of Composite Materials, Birkhäuser, Boston, to appear. | Zbl

[15] G. Dal Maso, Limits of minimum problems for general integralfunctionals with unilateral obstacles, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 74 (1983), 55-61. | Zbl

[16] G. Dal Maso, On the integral representation of certain local functionals, Ricerche Mat. 32 (1983), 85-113. | Zbl

[17] G. Dal Maso, r-convergence and μ-capacities,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 14 (1987), 423-464. | Numdam | Zbl

[18] G. Dal Maso, An Introduction to Γ-Convergence, Birkhäuser, Boston, 1993. | Zbl

[19] G. Dal Maso - A. Defranceschi, Some properties of a class of nonlinear variational μ-capacities, J. Funct. Anal. 79 (1988), 476-492. | Zbl

[20] G. Dal Maso - A. Defranceschi, Limits of nonlinear Dirichlet problems in varying domains, Manuscripta Math. 61 (1988), 251-278. | Zbl

[21] G. Dal Maso - A. Garroni, New results on the asymptotic behaviour of Dirichlet problems in perforated domains, Math. Mod. Meth. Appl. Sci. 3 (1994), 373-407. | Zbl

[22] G. Dal Maso - A. Malusa, Approximation of relaxed Dirichlet problems by boundary value problems in perforated domains, Proc. Roy. Soc. Edinburgh Sect. A 124 (1995), 99-114. | Zbl

[23] G. Dal Maso - U. Mosco, Wiener criteria and energy decay for relaxed Dirichlet problems, Arch. Rational Mech. Anal. 95 (1986), 345-387. | Zbl

[24] G. Dal Maso - U. Mosco, Wiener's criterion and r-convergence, Appl. Math. Optim. 15 (1987), 15-63. | Zbl

[25] G. Dal Maso - F. Murat, Dirichlet problems in perforated domains for homogeneous monotone operators on H10, Calculus of Variations, Homogenization and Continuum Mechanics (CIRM-Luminy, Marseille, 1993), 177-202, World Scientific, Singapore, 1994. | Zbl

[26] G. Dal Maso - I.V. Skrypnik, Asymptotic behaviour of nonlinear Dirichlet problems in perforated domains, Ann. Mat. Pura Appl., to appear. | Zbl

[27] A. Defranceschi - E. Vitali, Limits of minimum problems with convex obstacles for vector valued functions, Applicable Anal. 52 (1994), 1-33. | Zbl

[28] P.R. Halmos, Measure Theory, Van Nostrand, Princeton, 1950. | Zbl

[29] L.I. Hedberg, Non-linear potentials and approximation in the mean by analytic functions, Math. Z. 129 (1972), 299-319. | Zbl

[30] J. Heinonen - T. Kilpelxinen - O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Clarendon Press, Oxford, 1993. | Zbl

[31] E. Ya. Khruslov, The method of orthogonal projections and the Dirichlet problems in domains with a fine-grained boundary, Math. USSR-Sb. 17 (1972), 37-59. | Zbl

[32] A.A. Kovalevsky, On G-convergence of operators of Dirichlet problem with varying domain, Dokl. Akad. Nauk Ukrain. Ser. A 5 (1993), 13-17. | Zbl

[33] N. Labani - C. Picard, Homogenization of nonlinear Dirichlet problem in a periodically perforated domain, Recent Advances in Nonlinear Elliptic and Parabolic Problems (Nancy, 1988), 294-305, Res. Notes in Math. 208, Longman, Harlow, 1989. | Zbl

[34] O.A. Ladyzhenskaya - N.N. Ural'Tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968. | Zbl

[35] J.L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Gauthier-Villars, Paris, 1969. | Zbl

[36] A. Malusa - L. Orsina, Existence and regularity results for relaxed Dirichlet problems with measure data, Ann. Mat. Pura Appl., to appear. | Zbl

[37] A.V. Marchenko - E. Ya. Khruslov, Boundary Value Problems in Domains with Fine-Granulated Boundaries (in Russian), Naukova Dumka, Kiev, 1974. | Zbl

[38] T. Mekkaoui - C. Picard, Error estimate for the homogenization of a quasilinear Dirichlet problem in a periodically perfored domain, Progress in Partial Differential Equations: the Metz Survey 2, 185-193, Pitman Res. Notes in Math. 296, Longman, New York, 1993. | Zbl

[39] L. Notarantonio, Asymptotic behaviour of Dirichlet problems on a Riemannian manifold, Ricerche Mat. 41 (1992), 327-367. | Zbl

[40] I.V. Skrypnik, Nonlinear Elliptic Boundary Value Problems, Teubner-Verlag, Leipzig, 1986. | Zbl

[41] I.V. Skrypnik, Methods for Analysis of Nonlinear Elliptic Boundary Value Problems (in Russian), Nauka, Moscow, 1990. English translation in: Translations of Mathematical Monographs 139, American Mathematical Society, Providence, 1994. | Zbl

[42] I.V. Skrypnik, Averaging nonlinear Dirichlet problems in domains with channels, Soviet Math. Dokl. 42 (1991), 853-857. | Zbl

[43] I.V. Skrypnik, Asymptotic behaviour of solutions of nonlinear elliptic problems in perforated domains, Mat. Sb. (N.S.) 184 (1993), 67-90. | Zbl

[44] I.V. Skrypnik, Homogenization of nonlinear Dirichlet problems in perforated domains of general structure, Mat. Sb. (N.S.) 187 (1996), 125-157. | Zbl

[45] I.V. Skrypnik, New conditions for the homogenization of nonlinear Dirichlet problems in perforated domains, Ukrain. Mat. Zh., to appear. | Zbl

[46] L. Tartar, Cours Peccot au Collège de France Paris, 1977, partially written in: Murat F.: H-convergence. Séminaire d'Analyse Fonctionnelle et Numérique, Université d'Alger, 1977-78. English translation: Murat F., Tartar L.: H-convergence. Topics in the Mathematical Modelling of Composite Materials, Birkhäuser, Boston, to appear.

[47] W.P. Ziemer, Weakly Differentiable Functions, Springer-Verlag, Berlin, 1989. | Zbl