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Homogenisation of Dirichlet problems for monotone operators in varying domains

Published online by Cambridge University Press:  14 November 2011

Juan Casado-Díaz
Juan Casado-Díaz
Affiliation:
Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Fac. de Matemáticas, C. Tarfía s/n, 41012 Sevilla, Spain
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Abstract

We study the asymptotic behaviour, for a sequence of varying open sets Ωn, of the solutions un of nonlinear Dirichlet problems for a monotone Leray–Lions operator. The method is based on the comparison between the gradient of un and the corrector for the p-Laplacian corresponding to the same geometry as the monotone operator. The representation of the limit problem and a corrector result are obtained.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 127 , Issue 3 , 1997 , pp. 457 - 478
Copyright
Copyright © Royal Society of Edinburgh 1997

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References

1Attouch, H.. Variational Convergence for Functions and Operators (London: Pitman, 1984).Google Scholar
2Boccardo, L. and Murat, F.. Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations. Nonlinear Anal. 19 (1992), 581–97.CrossRefGoogle Scholar
3Casado-Dìaz, J.. Sobre la homogeneización de problemas no coercivos y problemas en dominios con agujeros(Ph.D. Thesis, University of Seville, 1993).Google Scholar
4Casado-Dìaz, J.. Existence of a sequence satisfying Cioranescu-Murat conditions in homogenization of Dirichlet problems in perforated domains. Rend. Mat. 16 (1996), 387–413.Google Scholar
5Casado-Dìaz, J.. Homogenization of general quasi-linear Dirichlet problems with quadratic growth in perforated domains. J. Math. Pur. Appl. (to appear).Google Scholar
6Casado-Dìaz, J.. Homogenization of a quasi-linear problem with quadratic growth in perforated domains: An example. Ann. Inst. H. Poincaré, Anal. Non Linéaire (to appear).Google Scholar
7Casado-Dìaz, J. and Garroni, A.. Asymptotic behaviour of Dirichlet solutions of nonlinear elliptic systems in varying domains (to appear).Google Scholar
8Cioranescu, D. and Murat, F.. Un terme étrange venu d'ailleurs. In Nonlinear partial differential equations and their applications. Collège de France seminar, Vols II and III, eds Brézis, H. and Lions, J.-L., Research Notes in Mathematics 60 and 70, 98–138 and 154–78 (London: Pitman, 1982).Google Scholar
9Dal Maso, G. and Defranceschi, A.. Limits of nonlinear Dirichlet problems in varying domains. Manuscripta Math. 61 (1988), 251–78.CrossRefGoogle Scholar
10Dal Maso, G. and Garroni, A.. New results on the asymptotic behaviour of Dirichlet problems in perforated domains. Math. Models Methods Appl. Sci. 3 (1994), 373–407.CrossRefGoogle Scholar
11Dal Maso, G. and Mosco, U.. Wiener-criteria and energy decay for relaxed Dirichlet problems. Arch. Rational Mech. Anal. 95 (1986), 345–87.CrossRefGoogle Scholar
12Dal Maso, G. and Mosco, U.. Wiener-criterion and Г-convergence. Appl. Math. Optim. 15 (1987), 15–63.CrossRefGoogle Scholar
13Dal Maso, G. and Murat, F.. Dirichlet problems in perforated domains for homogeneous monotone operators on H 10 In Calculus of variations, homogenization and continuum mechanics (Proceedings, Cirm-Lumigny, Marseille, June 21–25, 1993), eds Bouchitté, G., Buttazzo, G. and Suquet, P., Series on Advances in Mathematics for Applied Sciences 18, 177–202 (Singapore: World Scientific, 1994).Google Scholar
14Dal Maso, G. and Murat, F.. Asymptotic behaviour and correctors for Dirichlet problems in perforated domains with homogeneous monotone operators (to appear).Google Scholar
15Deny, J.. Les potentiels d'énergie finie. Acta Math. 82 (1950), 107–83.CrossRefGoogle Scholar
16Evans, L. C. and Gariepy, R. F.. Measure theory and fine properties of functions, Studies in Advanced Mathematics (Boca Raton: CRC Press, 1992).Google Scholar
17Federer, H. and Ziemer, W.. The Lebesgue set of a function whose distribution derivatives are p-th power summable. Indiana Univ. Math. J. 22 (1972), 139–58.CrossRefGoogle Scholar
18Finzi Vita, S. and Tchou, N. A.. Corrector results for relaxed Dirichlet problems. Asymptotic Anal. 5 (1992), 269–81.CrossRefGoogle Scholar
19Folland, G. B.. Real analysis. Modern techniques and their applications (New York: John Wiley, 1984).Google Scholar
20Kacimi, H. and Murat., F. Estimation de l'erreur dans des problèmes de Dirichlet où apparaît un terme étrange. In Partial differential equations and the calculus of variations, Vol. II, Essays in honor of De Giorgi, E., eds Colombini, F., Modica, L., Marino, A. and Spagnolo, S., Progress in Nonlinear Differential Equations and their Applications 2, 661–96 (Boston: Birkhäuser, 1989).Google Scholar
21Labani, N. and Picard, C.. Homogenization of a nonlinear Dirichlet problem in a periodically perforated domain. In Recent advances in nonlinear elliptic and parabolic problems, eds Benilan, P., Chipot, M., Evans, L. C. and Pierre, M., , Pitman Research Notes in Mathematics 208, 294–305 (Harlow: Longman, 1989).Google Scholar
22Skrypnik, I. V.. Nonlinear Elliptic Boundary Value Problems (Leipzig: Teubner, 1986).Google Scholar
23Skrypnik, I. V.. Asymptotic behaviour of solutions of nonlinear elliptic problems in perforated domains. Mat. Sb. 184(10) (1993), 67–90.Google Scholar
24Skrypnik, I. V.. Homogenization of nonlinear Dirichlet problems in perforated domains of general structure (to appear).Google Scholar
25ziemer, W. P.. Weakly Differentiable Functions (New York: Springer, 1989).CrossRefGoogle Scholar