×

Asymptotic behavior of a solution to a boundary value problem in a perforated domain with oscillating boundary. (English. Russian original) Zbl 0918.35043

Sib. Math. J. 39, No. 4, 621-644 (1998); translation from Sib. Mat. Zh. 39, No. 4, 730-754 (1998).
The aim of this article is to study a model problem for the Poisson equation in a perforated domain with very rapidly oscillating outer boundary in the presence of small dissipation on the boundaries of holes. This model appears in material technology for studying the macroscopic behavior of micro-inhomogeneous perforated media and bodies with rough surfaces. The authors study a particular case of such a medium in which perforation as well as oscillation of the boundary are locally periodic and their structures are assumed to be adjusted. In the study of locally periodic perforation, a difficulty arises: the geometry of cavities is not fixed. The method of compensated compactness or the method of two-scale convergence can be applied for constructing a limit problem, but it provides no estimates for the error. In order to overcome this obstacle, the authors use the technique of asymptotic expansion which requires the data to be regular but makes it possible to estimate the convergence rate. Namely, two terms of the interior asymptotic expansion guarantee an estimate of order \(\sqrt\varepsilon\) in the \(H^1\)-norm. The estimate for the residual is improved by constructing a boundary layer corrector that yields to an estimate of order \(\varepsilon\).

MSC:

35J25 Boundary value problems for second-order elliptic equations
35B40 Asymptotic behavior of solutions to PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35C20 Asymptotic expansions of solutions to PDEs
35A20 Analyticity in context of PDEs
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] V. A. Marchenko and E. Ya. Khruslov, Boundary Value Problems in Domains with a Fine-Grained Boundary [in Russian], Naukova Dumka, Kiev (1974). · Zbl 0289.35002
[2] E. Sanchez-Palencia, Homogenization Techniques for Composite Media, Springer-Verlag, Berlin and New York (1987). · Zbl 0619.00027
[3] G. A. Iosif’yan, O. A. Oleînik, and A. S. Shamaev, Mathematical Problems in the Theory of Strongly Inhomogeneous Elastic Media [in Russian], Moscow Univ., Moscow (1990). · Zbl 0746.73003
[4] V. V. Zhikov and S. M. Kozlov, and O. A. Oleînik, Averaging of Differential Operators [in Russian], Nauka, Moscow (1993). · Zbl 0487.35050
[5] D. Cioranescu and J. Saint Jean Paulin, ”Truss structures: Fourier conditions and eigenvalue problems,” in: Boundary Control and Boundary Variation, Springer-Verlag, Berlin and New York, 1992, pp. 125–141 (Lecture Notes in Control and Inform. Sci.,178). · Zbl 0779.93054
[6] D. Cioranescu and P. Donato, ”On a Robin problem in perforated domains,” in: Homogenization and Applications to Material Sciences, Gakkōtosho, Tokyo, 1997, pp. 123–136. (GAKUTO Internat. Ser. Math. Sci. Appl.,9. · Zbl 0900.35040
[7] D. Cioranescu and P. Donato, ”Homogénésation du problème de Neumann non homogène dans des ouverts perforés,” Asymptotic Anal.,1, No. 2, 115–138 (1988). · Zbl 0683.35026
[8] O. A. Oleinik and T. A. Shaposhnikova, ”On the homogenization of the Poisson equation in partially perforated domain with the arbitrary density of cavities and mixed conditions on their boundary,” Matematica e Applicazioni, Rendiconti Lincei Ser. IX,8, No. 3, 129–146 (1997). · Zbl 0878.35011
[9] E. Sanchez-Palencia and P. Suquet, ”Friction and homogenization of a boundary,” in: Free Boundary Problems. Theory and Applications, Pitman, London, 1983, pp. 561–571. · Zbl 0522.73093
[10] G. Bouchitte, A. Lidouh, and P. Suquet, ”Homogénéisation de frontière pour la modélisation du contact entre un corps déformable non linéaire et un corps rigide,” C. R. Acad. Sci. Paris Ser. I Math.,313, No. 13, 967–972 (1991). · Zbl 0751.73051
[11] G. Bouchitte, A. Lidouh, J. C. Michel, and P. Suquet, Might Boundary Homogenization Help to Understand Friction? International Centre for Theoretical Physics, Trieste (1993). SMR. 719/9.
[12] E. Sanchez-Palencia Inhomogeneous Media and Vibration Theory [Russian translation], Mir, Moscow (1984).
[13] A. G. Belyaev, ”Averaging the third boundary value problem for the Poisson equation in a domain with a rapidly oscillating boundary,” Vestnik Moskov. Univ. Ser. I Mat. Mekh., No. 6, 63–66 (1988). · Zbl 0691.35030
[14] A. G. Belyaev, On Singular Perturbations of Boundary Problems [in Russian], Dis. Kand. Fiz.-Mat. Nauk, Moscow Univ., Moscow (1990). · Zbl 0697.70016
[15] A. B. Movchan and S. A. Nazarov, ”The influence of small surface irregularities on the stress state of a body and the energy balance for a growing crack,” Prikl. Mat. Mekh.,55, No. 5, 819–828 (1991). · Zbl 0787.73058
[16] A. G. Belyaev, A. G. Mikheev, and A. S. Shamaev, ”Plane wave diffraction by a rapoidly oscillating surface,” Zh. Vychisl. Mat. i Mat. Fiz.,32, No. 8, 1258–1272 (1992). · Zbl 0778.35076
[17] G. A. Chechkin, A. Friedman, and A. L. Piatnitski, The Boundary Value Problem in Domains with Very Rapidly Oscillating Boundary, INRIA Rapport de Recherche No. 3062, Unité de Recherche–Institut National de Recherche en Informatique et en Automatique, Sophia Antipolis (1996). · Zbl 0938.35049
[18] F. Murat and L. Tartar, Calcul des Variations et Homogénéisation, Université Pierre et Marie Curie, Centre National de la Recherche Scientifique, Laboratoire d’Analyse Numérique, R 84012. Paris (1984).
[19] G. Allaire, ”Homogenization and two-scale convergence,” SIAM J. Math. Anal.,23, 1482–1518 (1992). · Zbl 0770.35005 · doi:10.1137/0523084
[20] N. S. Bakhvalov, ”Averaged characteristics of bodies with periodic structure,” Dokl. Akad. Nauk SSSR,218, No. 5, 1046–1048 (1974).
[21] N. S. Bakhvalov, ”Averaging of partial differential equations with rapidly oscillating coefficients,” Dokl. Akad. Nauk SSSR,221, No. 3, 516–519 (1975). · Zbl 0331.35009
[22] J.-L. Lions and E. Magenes, Inhomogeneous Boundary Value Problems and Their Applications [Russian translation], Mir, Moscow (1971). · Zbl 0212.43801
[23] S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics [in Russian], Nauka, Moscow (1988). · Zbl 0662.46001
[24] M. Briane, Homogénéisation de Matériaux Fibres et Multi-Couches, PhD Thesis, University Paris 6, Paris (1990).
[25] J. M. Ball and F. Murat, Remarks on Rank-One Convexity and Quasiconvexity, Université Pierre et Marie Curie, Centre National de la Recherche Scientifique, Laboratoire d’Analyse Numérique. Paris (1990). Report 90043.
[26] E. M. Landis and G. P. Panasenko, ”A theorem on the asymptotic behavior of the solutions of elliptic equations with coefficients that are periodic in all variables, except one,” Dokl. Akad. Nauk SSSR,235, No. 6, 1253–1255 (1977). · Zbl 0378.35020
[27] O. A. Oleînik and G. A. Iosif’yan, ”On the behavior at infinity of solutions to second-order elliptic equations in domains with noncompact boundary,” Mat. Sb.,112, No. 4, 588–610 (1980).
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.