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

Gianni Dal Maso; François Murat

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (1997)

  • Volume: 24, Issue: 2, page 239-290
  • ISSN: 0391-173X

How to cite

top

Dal Maso, Gianni, and Murat, François. "Asymptotic behaviour and correctors for Dirichlet problems in perforated domains with homogeneous monotone operators." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 24.2 (1997): 239-290. <http://eudml.org/doc/84259>.

@article{DalMaso1997,
author = {Dal Maso, Gianni, Murat, François},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
keywords = {monotone elliptic operators; relaxed Dirichlet problems; asymptotic behaviour},
language = {eng},
number = {2},
pages = {239-290},
publisher = {Scuola normale superiore},
title = {Asymptotic behaviour and correctors for Dirichlet problems in perforated domains with homogeneous monotone operators},
url = {http://eudml.org/doc/84259},
volume = {24},
year = {1997},
}

TY - JOUR
AU - Dal Maso, Gianni
AU - Murat, François
TI - Asymptotic behaviour and correctors for Dirichlet problems in perforated domains with homogeneous monotone operators
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 1997
PB - Scuola normale superiore
VL - 24
IS - 2
SP - 239
EP - 290
LA - eng
KW - monotone elliptic operators; relaxed Dirichlet problems; asymptotic behaviour
UR - http://eudml.org/doc/84259
ER -

References

top
  1. [1] H. Attouch, Variational Convergence for Functions and Operators., Pitman, London, 1984. Zbl0561.49012
  2. [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. [3] T. Bagby, Quasi topologies and rational approximation, J. Funct. Anal.10 (1972), 259-268. Zbl0266.30024
  4. [4] L. Boccardo - F. Murat, Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlinear Anal. 19 (1992), 581-597. Zbl0783.35020
  5. [5] G. Buttazzo - G. Dal Maso, Shape optimizationforDirichletproblems: relaxed solutions and optimality conditions, Appl. Math. Optim.23 (1991), 17-49. Zbl0762.49017
  6. [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. Zbl0622.28006
  7. [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. Zbl0699.35062
  8. [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. Zbl0942.35051
  9. [9] J. Casado Diaz, Homogenization of general quasi-linear problems with quadratic growth in perforated domains, J. Math. Pures Appl., to appear. Zbl0880.35015
  10. [10] J. Casado Diaz, Homogenization of Dirichlet problems for monotone operators in varying domains, Proc. Roy. Soc. Edinburgh Sect. A, to appear. Zbl0877.35013
  11. [11] J. Casado Diaz, Existence of a sequence satisfying Cioranescu-Murat conditions in homogenization of Dirichlet problems in perforated domains, Rend. Mat., to appear. Zbl0870.35013
  12. [12] J. Casado Diaz - A. Garroni, Asymptotic behaviour of nonlinear elliptic systems on varying domains, to appear. Zbl0952.35036
  13. [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. Zbl0769.35013
  14. [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. Zbl0496.35030
  15. [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. Zbl0526.49006
  16. [16] G. Dal Maso, On the integral representation of certain local functionals, Ricerche Mat. 32 (1983), 85-113. Zbl0543.49001
  17. [17] G. Dal Maso, r-convergence and μ-capacities,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 14 (1987), 423-464. Zbl0657.49005
  18. [18] G. Dal Maso, An Introduction to Γ-Convergence, Birkhäuser, Boston, 1993. Zbl0816.49001
  19. [19] G. Dal Maso - A. Defranceschi, Some properties of a class of nonlinear variational μ-capacities, J. Funct. Anal.79 (1988), 476-492. Zbl0657.49006
  20. [20] G. Dal Maso - A. Defranceschi, Limits of nonlinear Dirichlet problems in varying domains, Manuscripta Math. 61 (1988), 251-278. Zbl0653.49017
  21. [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. Zbl0804.47050
  22. [22] G. Dal Maso - A. Malusa, Approximation of relaxed Dirichlet problems by boundary value problems in perforated domains, Proc. Roy. Soc. Edinburgh Sect. A124 (1995), 99-114. Zbl0828.35007
  23. [23] G. Dal Maso - U. Mosco, Wiener criteria and energy decay for relaxed Dirichlet problems, Arch. Rational Mech. Anal. 95 (1986), 345-387. Zbl0634.35033
  24. [24] G. Dal Maso - U. Mosco, Wiener's criterion and r-convergence, Appl. Math. Optim. 15 (1987), 15-63. Zbl0644.35033
  25. [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. Zbl0884.47026
  26. [26] G. Dal Maso - I.V. Skrypnik, Asymptotic behaviour of nonlinear Dirichlet problems in perforated domains, Ann. Mat. Pura Appl., to appear. Zbl0991.35034
  27. [27] A. Defranceschi - E. Vitali, Limits of minimum problems with convex obstacles for vector valued functions, Applicable Anal. 52 (1994), 1-33. Zbl0839.49011
  28. [28] P.R. Halmos, Measure Theory, Van Nostrand, Princeton, 1950. Zbl0040.16802
  29. [29] L.I. Hedberg, Non-linear potentials and approximation in the mean by analytic functions, Math. Z.129 (1972), 299-319. Zbl0236.31010
  30. [30] J. Heinonen - T. Kilpelxinen - O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Clarendon Press, Oxford, 1993. Zbl0780.31001
  31. [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. Zbl0251.35032
  32. [32] A.A. Kovalevsky, On G-convergence of operators of Dirichlet problem with varying domain, Dokl. Akad. Nauk Ukrain. Ser. A5 (1993), 13-17. Zbl0817.47066
  33. [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. Zbl0782.35006
  34. [34] O.A. Ladyzhenskaya - N.N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968. Zbl0164.13002
  35. [35] J.L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Gauthier-Villars, Paris, 1969. Zbl0189.40603
  36. [36] A. Malusa - L. Orsina, Existence and regularity results for relaxed Dirichlet problems with measure data, Ann. Mat. Pura Appl., to appear. Zbl0882.35037
  37. [37] A.V. Marchenko - E. Ya. Khruslov, Boundary Value Problems in Domains with Fine-Granulated Boundaries (in Russian), Naukova Dumka, Kiev, 1974. Zbl0289.35002
  38. [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 Survey2, 185-193, Pitman Res. Notes in Math. 296, Longman, New York, 1993. Zbl0831.35015
  39. [39] L. Notarantonio, Asymptotic behaviour of Dirichlet problems on a Riemannian manifold, Ricerche Mat. 41 (1992), 327-367. Zbl0802.58058
  40. [40] I.V. Skrypnik, Nonlinear Elliptic Boundary Value Problems, Teubner-Verlag, Leipzig, 1986. Zbl0617.35001
  41. [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. Zbl0822.35001
  42. [42] I.V. Skrypnik, Averaging nonlinear Dirichlet problems in domains with channels, Soviet Math. Dokl. 42 (1991), 853-857. Zbl0757.35027
  43. [43] I.V. Skrypnik, Asymptotic behaviour of solutions of nonlinear elliptic problems in perforated domains, Mat. Sb. (N.S.)184 (1993), 67-90. Zbl0832.35053
  44. [44] I.V. Skrypnik, Homogenization of nonlinear Dirichlet problems in perforated domains of general structure, Mat. Sb. (N.S.)187 (1996), 125-157. Zbl0874.35012
  45. [45] I.V. Skrypnik, New conditions for the homogenization of nonlinear Dirichlet problems in perforated domains, Ukrain. Mat. Zh., to appear. Zbl0887.35015
  46. [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. [47] W.P. Ziemer, Weakly Differentiable Functions, Springer-Verlag, Berlin, 1989. Zbl0692.46022

Citations in EuDML Documents

top
  1. Mamadou Sango, Homogenization of the Dirichlet problem for a system of quasilinear elliptic equations in a domain with fine-grained boundary
  2. Carlos Conca, François Murat, Claudia Timofte, A generalized strange term in Signorini’s type problems
  3. B. Amaziane, S. Antontsev, L. Pankratov, A. Piatnitski, Homogenization of p -laplacian in perforated domain
  4. Carmen Calvo-Jurado, Juan Casado-Díaz, Manuel Luna-Laynez, Asymptotic behavior of nonlinear systems in varying domains with boundary conditions on varying sets
  5. Gianni Dal Maso, François Murat, Asymptotic behaviour and correctors for linear Dirichlet problems with simultaneously varying operators and domains
  6. Ciro D'Apice, Umberto De Maio, Peter I. Kogut, Suboptimal boundary controls for elliptic equation in critically perforated domain
  7. Carlos Conca, François Murat, Claudia Timofte, A Generalized Strange Term in Signorini's Type Problems

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.