Comptes Rendus
no. 1
Asymptotics at infinity of solutions to the Neumann problem in a sieve-type layer

Presented by: Évariste Sanchez-Palencia

Sergueı̈ A. Nazarov1; Gudrun Thäter2
1 Inst. of Mechanical Engineering Problems, V. O. Bol'shoı̈ pr. 61, St. Petersburg, 199178, Russia
2 Institut für Angewandte Mathematik, Universität Hannover, Welfengarten 1, 30167 Hannover, Germany
Comptes Rendus. Mécanique, Volume 331 (2003) no. 1, pp. 85-90.

The Neumann problem is considered in a domain Ω, which can differ from a periodic layer inside a compact set. We prove the Fredholm property of the corresponding operator in step-weighted Sobolev spaces and determine its kernel and cokernel. All these results are based on the obtained asymptotic representation of solutions at infinity.

On considère un problème de Neumann dans un domaine Ω, qui coincide avec une couche périodique excepté sur une partie compacte. On démontre que l'opérateur correspondant satisfait à la propriété de Fredholm dans des espaces de Sobolev avec poids et on détermine son noyau et son conoyau. Tous ces résultats sont déduits de la représentation asymptotique des solutions à l'infini.

Received:
Accepted:
Published online:
DOI: 10.1016/S1631-0721(02)00005-0
Keywords: Computational solid mechanics, Periodic layer, Homogenization procedure, Asymptotic behaviour, Step-weighted spaces
Mots-clés : Mécanique des solides numérique, Couche périodique, Processus d'homogénéisation, Comportement asymptotique, Espaces avec poids

Sergueı̈ A. Nazarov 1; Gudrun Thäter 2

1 Inst. of Mechanical Engineering Problems, V. O. Bol'shoı̈ pr. 61, St. Petersburg, 199178, Russia
2 Institut für Angewandte Mathematik, Universität Hannover, Welfengarten 1, 30167 Hannover, Germany 
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Sergueı̈ A. Nazarov; Gudrun Thäter. Asymptotics at infinity of solutions to the Neumann problem in a sieve-type layer. Comptes Rendus. Mécanique, Volume 331 (2003) no. 1, pp. 85-90. doi : 10.1016/S1631-0721(02)00005-0. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/S1631-0721(02)00005-0/

[1] P.G. Ciarlet Plates and Junctions in Elastic Multi-Structures. An Asymptotic Analysis, Masson, Paris, 1990

[2] D. Leguillon; É. Sanchez-Palencia Approximation of a two dimensional problem of junctions, Comput. Mech., Volume 6 (1990), pp. 435-455

[3] S.A. Nazarov; B.A. Plamenevski Asymptotics of the spectrum of the Neumann problem in singularly perturbed thin domains, Leningr. Math. J., Volume 2 (1991), pp. 287-311

[4] V.A. Kozlov; V.G. Mazya; A.B. Movchan Asymptotic analysis of a mixed boundary value problem in a multi-structure, Asymptotic Anal., Volume 8 (1994), pp. 105-143

[5] S.A. Nazarov Junctions of singularly degenerating domains with different limit dimensions. 1 & 2, Trudy Sem. Petrovsk., Volume 18 (1995), pp. 3-78 and 20 (1997) 155–195

[6] S.A. Nazarov Asymptotics of the solution to a boundary value problem in a thin cylinder with nonsmooth lateral surface, Math. Izvestiya, Volume 42 (1994), pp. 183-217

[7] S.A. Nazarov Asymptotic expansions at infinity of solutions to the elasticity theory problem in a layer, Trudy Moskov. Mat. Obschch., Volume 60 (1998), pp. 3-97

[8] S.A. Nazarov; K.I. Pileckas The asymptotic properties of the solution to the Stokes problem in domains that are layer-like at infinity, J. Math. Fluid Mech., Volume 1 (1999), pp. 131-167

[9] S.A. Nazarov Asymptotics at infinity of the solution to the Dirichlet problem for a system of equations with periodic coefficients in an angular domain, Russian J. Math. Phys., Volume 3 (1995), pp. 297-326

[10] É. Sanchez-Palencia Non-Homogeneous Media and Vibration Theory, Lecture Notes in Phys., 127, Springer-Verlag, Berlin, 1980

[11] N.S. Bakhvalov; G. Panasenko Homogenisation: Averaging Processes in Periodic Media. Mathematical Problems in the Mechanics of Composite Materials, Kluwer Academic, Dordrecht, 1989

[12] S.A. Nazarov Vishik–Lyusternik method for elliptic boundary-value problems in regions with conical points. 1. The problem in a cone, Siberian Math. J., Volume 22 (1982), pp. 594-611

[13] V.A. Kondratiev Boundary problems for elliptic equations in domains with conical or angular points, Trans. Moscow Math. Soc., Volume 16 (1967), pp. 227-313

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