Spectral analysis in a thin domain with periodically oscillating characteristics

Rita Ferreira; Luísa M. Mascarenhas; Andrey Piatnitski

ESAIM: Control, Optimisation and Calculus of Variations (2012)

  • Volume: 18, Issue: 2, page 427-451
  • ISSN: 1292-8119

Abstract

top
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.

How to cite

top

Ferreira, Rita, Mascarenhas, Luísa M., and Piatnitski, Andrey. "Spectral analysis in a thin domain with periodically oscillating characteristics." ESAIM: Control, Optimisation and Calculus of Variations 18.2 (2012): 427-451. <http://eudml.org/doc/272885>.

@article{Ferreira2012,
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 &lt; 1), or ε is much greater than δ(δ = ετ, τ &gt; 1). We consider all three cases.},
author = {Ferreira, Rita, Mascarenhas, Luísa M., Piatnitski, Andrey},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {spectral analysis; dimension reduction; periodic homogenization; Γ-convergence; asymptotic expansions; -convergence},
language = {eng},
number = {2},
pages = {427-451},
publisher = {EDP-Sciences},
title = {Spectral analysis in a thin domain with periodically oscillating characteristics},
url = {http://eudml.org/doc/272885},
volume = {18},
year = {2012},
}

TY - JOUR
AU - Ferreira, Rita
AU - Mascarenhas, Luísa M.
AU - Piatnitski, Andrey
TI - Spectral analysis in a thin domain with periodically oscillating characteristics
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2012
PB - EDP-Sciences
VL - 18
IS - 2
SP - 427
EP - 451
AB - 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 &lt; 1), or ε is much greater than δ(δ = ετ, τ &gt; 1). We consider all three cases.
LA - eng
KW - spectral analysis; dimension reduction; periodic homogenization; Γ-convergence; asymptotic expansions; -convergence
UR - http://eudml.org/doc/272885
ER -

References

top
  1. [1] G. Allaire and C. Conca, Bloch wave homogenization and spectral asymptotic analysis. J. Math. Pures Appl.77 (1998) 153–208. Zbl0901.35005MR1614641
  2. [2] G. Allaire and F. Malige, Analyse asymptotique spectrale d’un problème de diffusion neutronique. C. R. Acad. Sci. Paris, Ser. I 324 (1997) 939–944. Zbl0879.35153MR1450451
  3. [3] H. Attouch, Variational convergence for functions and operators. Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA (1984). Zbl0561.49012MR773850
  4. [4] N.S. Bachvalov and G.P. Panasenko, Homogenization of Processes in Periodic Media. Nauka, Moscow (1984). Zbl0692.73012MR797571
  5. [5] A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic analysis for periodic structures. North-Holland Publishing Co., Amsterdam (1978). Zbl1229.35001MR503330
  6. [6] G. Bouchitté, M.L. Mascarenhas and L. Trabucho, On the curvature and torsion effects in one dimensional waveguides. ESAIM : COCV 13 (2007) 793–808. Zbl1139.49043MR2351404
  7. [7] G. Dal Maso, An introduction to Γ-convergence, Progress in Nonlinear Differential Equations and their Applications 8. Birkhäuser Boston Inc., Boston (1993). Zbl0816.49001MR1201152
  8. [8] R. Ferreira and M.L. Mascarenhas, Waves in a thin and periodically oscillating medium. C. R. Math. Acad. Sci. Paris, Ser. I 346 (2008) 579–584. Zbl1147.35065MR2412801
  9. [9] D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order. Classics in Mathematics, Springer-Verlag, New York (1977). Zbl1042.35002MR473443
  10. [10] V. Jikov, S. Kozlov and O. Oleĭnik, Homogenization of differential operators and integral functionals. Springer-Verlag, Berlin (1994). Zbl0801.35001MR1329546
  11. [11] S. Kesavan, Homogenization of Elliptic Eigenvalue Problems : Part 1. Appl. Math. Optim.5 (1979) 153–167. Zbl0415.35061MR533617
  12. [12] S. Kesavan, Homogenization of Elliptic Eigenvalue Problems : Part 2. Appl. Math. Optim.5 (1979) 197–216. Zbl0428.35062
  13. [13] S. Kozlov and A. Piatnitski, Effective diffusion for a parabolic operator with periodic potential. SIAM J. Appl. Math.53 (1993) 401–418. Zbl0805.35006MR1212756
  14. [14] S. Kozlov and A. Piatnitski, Degeneration of effective diffusion in the presence of periodic potential. Ann. Inst. H. Poincaré Probab. Statist.32 (1996) 571–587. Zbl0888.35013MR1411272
  15. [15] 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). Zbl0920.35019MR1493039
  16. [16] 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). Zbl0768.73003MR1195131
  17. [17] M. Vanninathan, Homogenization of eigenvalue problems in perforated domains. Proc. Indian Acad. Sci. Math Sci.90 (1981) 239–271. Zbl0486.35063MR635561
  18. [18] M.I. Vishik and L.A. Lyusternik, Regular degeneration and boundary layer for linear differential equations with small parameter. Amer. Math. Soc. Transl. (2) 20 (1962) 239–364 [English translation]. Zbl0122.32402MR136861

NotesEmbed ?

top

You must be logged in to post comments.