Hostname: page-component-76fb5796d-vfjqv Total loading time: 0 Render date: 2024-04-26T13:44:09.623Z Has data issue: false hasContentIssue false
  • English
  • Français

Which sequences of holes are admissible for periodic homogenization with Neumann boundary condition?

Published online by Cambridge University Press:  15 August 2002

Alain Damlamian  and
Patrizia Donato
Alain Damlamian
Affiliation:
Laboratoire d'Analyse et de Mathématiques Appliquée, UMR 8050 du CNRS, Universités de Marne-la-Vallée et Paris 12 Val-de-Marne, Université Paris 12, 94010 Créteil Cedex France; damlamian@univ-paris12.fr.
Patrizia Donato
Affiliation:
Laboratoire de Mathématiques Raphaël Salem, Université de Rouen, Site Colbert, 76821 Mont-Saint-Aignan Cedex, France; donato@univ-rouen.fr. Université Paris VI, Laboratoire Jacques-Louis Lions, Boîte Courrier 187, 75252 Paris Cedex, France; donato@ann.jussieu.fr.
Get access

Abstract

In this paper we give a general presentation of the homogenization of Neumann type problems in periodically perforated domains, including the case where the shape of the reference hole varies with the size of the period (in the spirit of the construction of self-similar fractals). We shows that H0-convergence holds under the extra assumption that there exists a bounded sequence of extension operators for the reference holes. The general class of Jones-domains gives an example where this result applies. When this assumption fails, another approach, using the Poincaré–Wirtinger inequality is presented. A corresponding class where it applies is that of John-domains, for which the Poincaré–Wirtinger constant is controlled. The relationship between these two kinds of assumptions is also clarified.

Keywords

Periodic homogenizationperforated domainsH0-convergencePoincaré–Wirtinger inequalityJones domainsJohn domains.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 8: A tribute to JL Lions , 2002 , pp. 555 - 585
Copyright
© EDP Sciences, SMAI, 2002

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Acerbi, E., Chiado' Piat, V., Dal Maso, G. and Percivale, D., An extension theorem for connected sets, and homogenization in general periodic domains. Nonlinear Anal. TMA 18 (1992) 481-495. CrossRef
Allaire, G. and Murat, F., Homogenization of the Neumann problem with non-isolated holes. Asymptot. Anal. 7 (1993) 81-95.
H. Attouch, Variational convergence for functions and operators. Pitman, Boston, Appl. Math. Ser. (1984).
N.S. Bakhvalov and G.P. Panasenko, Homogenization: Averaging Processes in Periodic Media. Kluwer, Dordrecht (1989).
A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures. North-Holland, Amsterdam (1978).
B. Bojarski, Remarks on Sobolev imbedding inequalities, in Complex Analysis. Springer-Verlag, Lecture Notes in Math. 1351 (1988) 257-324.
Briane, M., Poincare'-Wirtinger's inequality for the homogenization in perforated domains. Boll. Un. Mat. Ital. B 11 (1997) 53-82.
M. Briane, A. Damlamian and P. Donato, H-convergence in perforated domains, in Nonlinear Partial Differential Equations Appl., Collège de France Seminar, Vol. XIII, edited by D. Cioranescu and J.-L. Lions. Longman, New York, Pitman Res. Notes in Math. Ser. 391 (1998) 62-100.
Buckley, S. and Koskela, P., Sobolev-Poincaré implies John. Math. Res. Lett. 2 (1995) 577-593. CrossRef
Chenais, D., On the existence of a solution in a domain identification problem. J. Math. Anal. Appl. 52 (1975) 189-219. CrossRef
D. Cioranescu and P. Donato, An Introduction to Homogenization. Oxford University Press, Oxford Lecture Ser. in Math. Appl. 17 (1999).
Cioranescu, D. and Saint Jean Paulin, J., Homogenization in open sets with holes. J. Math. Anal. Appl. 71 (1979) 590-607. CrossRef
D. Cioranescu and J. Saint Jean Paulin, Homogenization of reticulated structures. Springer-Verlag, Berlin, New York (1999).
Conca, C. and Donato, P., Non-homogeneous Neumann problems in domains with small holes. ESAIM: M2AN 22 (1988) 561-608. CrossRef
Damlamian, A. and Donato, P., Homogenization with small shape-varying perforations. SIAM J. Math. Anal. 22 (1991) 639-652. CrossRef
Gehring, F.W. and Martio, O., Quasiextremal distance domains and extension of quasiconformal mappings. J. Anal. Math. 45 (1985) 181-206. CrossRef
Gehring, F.W. and Osgood, B.G., Uniform domains and the quasi-hyperbolic metric. J. Anal. Math. 36 (1979) 50-74. CrossRef
E. Hruslov, The asymptotic behavior of solutions of the second boundary value problem under fragmentation of the boundary of the domain. Maths. USSR Sbornik 35 (1979).
Jones, P., Quasiconformal mappings and extensions of functions in Sobolev spaces. Acta Math. 1-2 (1981) 71-88. CrossRef
Martio, O., Definitions for uniform domains. Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980) 179-205. CrossRef
Martio, O., John domains, bilipschitz balls and Poincaré inequality. Rev. Roumaine Math. Pures Appl. 33 (1988) 107-112.
Martio, O. and Sarvas, J., Injectivity theorems in plane and space. Ann. Acad. Sci. Fenn. Ser. A I Math. 4 (1979) 383-401. CrossRef
V.G. Maz'ja, Sobolev spaces. Springer-Verlag, Berlin (1985).
F. Murat, H-Convergence, Séminaire d'Analyse Fonctionnelle et Numérique (1977/1978). Université d'Alger, Multigraphed.
F. Murat and L. Tartar, H-Convergence, in Topics in the Mathematical Modelling of Composite Materials, edited by A. Cherkaev and R. Kohn. Birkhäuser, Boston (1997) 21-43.
E. Sanchez-Palencia, Non homogeneous Media and Vibration Theory. Springer-Verlag, Lecture Notes in Phys. 127 (1980).
Smith, W. and Stegenga, D.A., Hölder domains and Poincaré domains. Trans. Amer. Math. Soc. 319 (1990) 67-100.
Spagnolo, S., Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche. Ann. Sc. Norm. Sup. Pisa 22 (1968) 571-597.
E.M. Stein, Singular integrals and differentiability properties of functions. Princeton University Press, Princeton, N.J. (1970).
L. Tartar, Cours Peccot au Collège de France (1977).
Väisalä, J., Uniform domains. Tohoku Math. J. 40 (1988) 101-118. CrossRef
Wallin, H., The trace to the boundary of Sobolev spaces on a snowflake. Manuscripta Math. 73 (1991) 117-125. CrossRef
Zhikov, V.V., Connectedness and Homogenization. Examples of fractal conductivity. Sbornik Math. 187 (1196) 1109-1147. CrossRef