Hostname: page-component-7b7567c8c-wxvwp Total loading time: 0 Render date: 2026-11-15T09:33:45.350Z Has data issue: false hasContentIssue false
  • English
  • Français

Spectral analysis in a thin domain with periodicallyoscillating characteristics

Published online by Cambridge University Press:  22 June 2011

Rita Ferreira,
Luísa M. Mascarenhas and
Andrey Piatnitski
Rita Ferreira
Affiliation:
I.C.T.I. - Carnegie Mellon | Portugal, F.C.T./C.M.A. da U.N.L., Quinta da Torre, 2829–516 Caparica, Portugal. rferreir@andrew.cmu.edu; ragf@fct.unl.pt
Luísa M. Mascarenhas
Affiliation:
Departamento de Matemática da F.C.T./C.M.A. da U.N.L., Quinta da Torre, 2829–516 Caparica, Portugal; mascar@fct.unl.pt
Andrey Piatnitski
Affiliation:
Narvik University College, P.O. Box 385, 8505 Narvik, NorwayP.N. Lebedev Physical Institute RAS, Leninski prospect 53, Moscow 119991, Russia; andrey@sci.lebedev.ru

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

The paper deals with a Dirichlet spectral problem for an elliptic operator withε-periodic coefficients in a 3D bounded domain of small thicknessδ. We study the asymptotic behavior of the spectrum asε and δ tend to zero. This asymptotic behavior dependscrucially on whether ε and δ are of the same order(δ ≈ ε), or ε is much less thanδ(δ = ετ, τ < 1),or ε is much greater thanδ(δ = ετ, τ > 1).We consider all three cases.

Keywords

Spectral analysisdimension reductionperiodic homogenizationΓ-convergenceasymptotic expansions

Information

Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 18 , Issue 2 , April 2012, pp. 427 - 451
Copyright
© EDP Sciences, SMAI, 2011

References

Allaire, G. and Conca, C., Bloch wave homogenization and spectral asymptotic analysis. J. Math. Pures Appl. 77 (1998) 153–208. Google Scholar
Allaire, G. and Malige, F., Analyse asymptotique spectrale d’un problème de diffusion neutronique. C. R. Acad. Sci. Paris, Ser. I 324 (1997) 939–944. Google Scholar
H. Attouch, Variational convergence for functions and operators. Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA (1984).
N.S. Bachvalov and G.P. Panasenko, Homogenization of Processes in Periodic Media. Nauka, Moscow (1984).
A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic analysis for periodic structures. North-Holland Publishing Co., Amsterdam (1978).
Bouchitté, G., Mascarenhas, M.L. and Trabucho, L., On the curvature and torsion effects in one dimensional waveguides. ESAIM : COCV 13 (2007) 793–808. Google Scholar
G. Dal Maso, An introduction to Γ -convergence, Progress in Nonlinear Differential Equations and their Applications 8. Birkhäuser Boston Inc., Boston (1993).
Ferreira, R. and Mascarenhas, M.L., Waves in a thin and periodically oscillating medium. C. R. Math. Acad. Sci. Paris, Ser. I 346 (2008) 579–584. Google Scholar
D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order. Classics in Mathematics, Springer-Verlag, New York (1977).
V. Jikov, S. Kozlov and O. Oleĭnik, Homogenization of differential operators and integral functionals. Springer-Verlag, Berlin (1994).
Kesavan, S., Homogenization of Elliptic Eigenvalue Problems : Part 1. Appl. Math. Optim. 5 (1979) 153–167. Google Scholar
Kesavan, S., Homogenization of Elliptic Eigenvalue Problems : Part 2. Appl. Math. Optim. 5 (1979) 197–216. Google Scholar
Kozlov, S. and Piatnitski, A., Effective diffusion for a parabolic operator with periodic potential. SIAM J. Appl. Math. 53 (1993) 401–418. Google Scholar
Kozlov, S. and Piatnitski, A., Degeneration of effective diffusion in the presence of periodic potential. Ann. Inst. H. Poincaré Probab. Statist. 32 (1996) 571–587. Google Scholar
F. Murat and L. Tartar, H-Convergence, in Topics in the mathematical modelling of composite materials. Progr. Nonlinear Differential Equations Appl., Birkhäuser Boston, Boston (1997).
O.A. Oleĭnik, A.S. Shamaev and G.A. Yosifian, Mathematical problems in elasticity and homogenization. Studies in Mathematics and its Applications, North-Holland Publishing Co., Amsterdam (1992).
Vanninathan, M., Homogenization of eigenvalue problems in perforated domains. Proc. Indian Acad. Sci. Math Sci. 90 (1981) 239–271. Google Scholar
Vishik, M.I. and Lyusternik, L.A., Regular degeneration and boundary layer for linear differential equations with small parameter. Amer. Math. Soc. Transl. (2) 20 (1962) 239–364 [English translation]. Google Scholar