# 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

## Access Full Article

top## Abstract

top## How to cite

topBouchitté, 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

top- G. Allaire and C. Conca, Bloch wave homogenization and spectral asymptotic analysis. J. Math. Pures Appl.77 (1998) 153–208.
- B. Chenaud, P. Duclos, P. Freitas and D. Krejčiřík, Geometrically induced discrete specrtum in curved tubes. Differ. Geometry Appl.23 (2005) 95–105.
- C. Conca, J. Planchard and M. Vanninathan, Fluids and periodic structures, Research in Applied Mathematics38. Masson, Paris (1995).
- G. Dal Maso, An Introduction to Γ -Convergence. Birkhäuser, Boston (1993).
- P. Duclos and P. Exner, Curvature-induced bounds states in quantum waveguides in two and tree dimensions. Rev. Math. Phys.7 (1995) 73–102.
- V. Jikov, S.M. Kozlov and O.A. Oleinik, Homogenization of Differential Operators and Integral Equations. Springer-Verlag, Berlin (1994).
- 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.
- 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.
- M. Vanninathan, Homogenization of eigenvalue problems in perforated domains. Proc. Indian Acad. Sci. Math. Sci.90 (1981) 239–271.

## Citations in EuDML Documents

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

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.