On the curvature and torsion effects in one dimensional waveguides

Guy Bouchitté; M. Luísa Mascarenhas; Luís Trabucho

ESAIM: Control, Optimisation and Calculus of Variations (2007)

  • Volume: 13, Issue: 4, page 793-808
  • ISSN: 1292-8119

Abstract

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We consider the Laplace operator in a thin tube of 3 with a Dirichlet condition on its boundary. We study asymptotically the spectrum of such an operator as the thickness of the tube's cross section goes to zero. In particular we analyse how the energy levels depend simultaneously on the curvature of the tube's central axis and on the rotation of the cross section with respect to the Frenet frame. The main argument is a Γ-convergence theorem for a suitable sequence of quadratic energies.

How to cite

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Bouchitté, Guy, Mascarenhas, M. Luísa, and Trabucho, Luís. "On the curvature and torsion effects in one dimensional waveguides." ESAIM: Control, Optimisation and Calculus of Variations 13.4 (2007): 793-808. <http://eudml.org/doc/250014>.

@article{Bouchitté2007,
abstract = { We consider the Laplace operator in a thin tube of $\{\mathbb R\}^3$ with a Dirichlet condition on its boundary. We study asymptotically the spectrum of such an operator as the thickness of the tube's cross section goes to zero. In particular we analyse how the energy levels depend simultaneously on the curvature of the tube's central axis and on the rotation of the cross section with respect to the Frenet frame. The main argument is a Γ-convergence theorem for a suitable sequence of quadratic energies. },
author = {Bouchitté, Guy, Mascarenhas, M. Luísa, Trabucho, Luís},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Dimension reduction; Γ-convergence; curvature and torsion; waveguides; dimension reduction; -convergence},
language = {eng},
month = {9},
number = {4},
pages = {793-808},
publisher = {EDP Sciences},
title = {On the curvature and torsion effects in one dimensional waveguides},
url = {http://eudml.org/doc/250014},
volume = {13},
year = {2007},
}

TY - JOUR
AU - Bouchitté, Guy
AU - Mascarenhas, M. Luísa
AU - Trabucho, Luís
TI - On the curvature and torsion effects in one dimensional waveguides
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2007/9//
PB - EDP Sciences
VL - 13
IS - 4
SP - 793
EP - 808
AB - We consider the Laplace operator in a thin tube of ${\mathbb R}^3$ with a Dirichlet condition on its boundary. We study asymptotically the spectrum of such an operator as the thickness of the tube's cross section goes to zero. In particular we analyse how the energy levels depend simultaneously on the curvature of the tube's central axis and on the rotation of the cross section with respect to the Frenet frame. The main argument is a Γ-convergence theorem for a suitable sequence of quadratic energies.
LA - eng
KW - Dimension reduction; Γ-convergence; curvature and torsion; waveguides; dimension reduction; -convergence
UR - http://eudml.org/doc/250014
ER -

References

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  1. G. Allaire and C. Conca, Bloch wave homogenization and spectral asymptotic analysis. J. Math. Pures Appl.77 (1998) 153–208.  Zbl0901.35005
  2. B. Chenaud, P. Duclos, P. Freitas and D. Krejčiřík, Geometrically induced discrete specrtum in curved tubes. Differ. Geometry Appl.23 (2005) 95–105.  Zbl1078.81022
  3. C. Conca, J. Planchard and M. Vanninathan, Fluids and periodic structures, Research in Applied Mathematics38. Masson, Paris (1995).  Zbl0910.76002
  4. G. Dal Maso, An Introduction to Γ -Convergence. Birkhäuser, Boston (1993).  
  5. P. Duclos and P. Exner, Curvature-induced bounds states in quantum waveguides in two and tree dimensions. Rev. Math. Phys.7 (1995) 73–102.  Zbl0837.35037
  6. V. Jikov, S.M. Kozlov and O.A. Oleinik, Homogenization of Differential Operators and Integral Equations. Springer-Verlag, Berlin (1994).  
  7. P. Kuchment, On some spectral problems of mathematical physics. Partial differential equations and inverse problems., Contemp. Math.362. Amer. Math. Soc., Providence, RI (2004) 241–276.  Zbl1061.35065
  8. J. Rubinstein, M. Schatzman, Variational problems on multiply connected thin strips. II. Convergence of the Ginzburg-Landau functional. Arch. Ration. Mech. Anal.160 (2001) 309–324.  Zbl0997.49004
  9. M. Vanninathan, Homogenization of eigenvalue problems in perforated domains. Proc. Indian Acad. Sci. Math. Sci.90 (1981) 239–271.  Zbl0486.35063

Citations in EuDML Documents

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  1. Rita Ferreira, Luísa M. Mascarenhas, Andrey Piatnitski, Spectral analysis in a thin domain with periodically oscillating characteristics
  2. Rita Ferreira, Luísa M. Mascarenhas, Andrey Piatnitski, Spectral analysis in a thin domain with periodically oscillating characteristics
  3. David Krejčiřík, Spectrum of the Laplacian in a narrow curved strip with combined Dirichlet and Neumann boundary conditions
  4. Denis Borisov, Giuseppe Cardone, Complete asymptotic expansions for eigenvalues of Dirichlet laplacian in thin three-dimensional rods
  5. David Krejčiřík, Spectrum of the laplacian in a narrow curved strip with combined Dirichlet and Neumann boundary conditions
  6. Denis Borisov, Giuseppe Cardone, Complete asymptotic expansions for eigenvalues of Dirichlet Laplacian in thin three-dimensional rods

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