Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-16T01:19:29.775Z Has data issue: false hasContentIssue false
  • English
  • Français

Spectral analysis in a thin domain with periodically oscillating 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. [email protected]; [email protected]
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; [email protected]
Andrey Piatnitski
Affiliation:
Narvik University College, P.O. Box 385, 8505 Narvik, Norway P.N. Lebedev Physical Institute RAS, Leninski prospect 53, Moscow 119991, Russia; [email protected]
Get access

Abstract

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

Keywords

Spectral analysisdimension reductionperiodic homogenizationΓ-convergenceasymptotic expansions
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

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

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