Abstract
The paper deals with the convergence, as ε tends to zero, of the spectrum of the Neumann problem -Δυɛ=λ(ɛ)υɛ in a «weakly connected» periodic domain Ωɛ of ℝ3. The domain Ωɛ is composed of a finite number of disjoint connected domains linked by thin bridges (curved plates or tubes). Under a few assumptions on the characteristic sizes of these bridges, we give an explicit asymptotic formula for the eigenvalues which tend to zero and we prove that the rest of the spectrum converges to the spectrum of an elliptic coupled system.
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E. Acerbi—V. Chiadò Piat—G. Dal Maso—D. Percivale,An extension theorem from connected sets, and homogenization in general periodic domains, Nonlinear Analysis, Theory Methods and Applications, Vol.18, No. 5 (1992), pp. 481–495.
G. Allaire—F. Murat,Homogenization of the Neumann problem with nonisolated holes, Asymptotic Analysis,7 (1993), pp. 81–95.
M. Briane,Homogenization in some weakly connected domains, Ricerche di Matematica,XLVII, no. 1 (1998), pp. 51–94.
D. Cioranescu—J. Saint Jean Paulin,Homogenization in open sets with holes, J. Math. Anal. Appl.,71 (1979), pp. 590–607.
E. Ya. Hruslov,The asymptotic behavior, of solutions of the second boundary value problems under fragmentation of the boundary of the domain, Maths. USSR Sbornik,35 no. 2 (1979).
F. Murat,H-convergence, Séminaire d'Analyse Fonctionnelle et Numérique, 1977–78, Université d'Alger. English translation inF. Murat—L. Tartar,H-convergence, Topics in the Mathematical Modelling of Composite Materials,A. Cherkaev—R. Kohn (eds.),Progress in Nonlinear Differential Equations and their Applications,31, Birkaüser, Boston (1997), pp. 21–43.
L. Tartar,Cours Peccot, Collège de France, 1997, unpublished, partially written in [6].F. Murat H-convergence, Séminaire d'Analyse Fonctionnelle et Numérique, 1977–78, Université d'Alger. English translation inF. Murat—L. Tartar H-convergence, Topics in the Mathematical Modelling of Composite Materials,A. Cherkaev—R. Kohn (eds.),Progress in Nonlinear Differential Equations and their Applications,31, Birkaüser, Boston (1997), pp. 21–43.
M. Vanninathan,Sur quelques problèmes d'homogénéisation dans les équations aux dérivées partielles, Thèse de Doctorat d'État ès Sciences de l'Université Paris 6 (1979).
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Entrata in Redazione il 13 gennaio 1997.
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Briane, M. The convergence of the spectrum of a weakly connected domain. Annali di Matematica pura ed applicata 177, 1–35 (1999). https://doi.org/10.1007/BF02505904
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DOI: https://doi.org/10.1007/BF02505904