We consider the Laplace operator in a thin tube of 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 -convergence theorem for a suitable sequence of quadratic energies.
Mots clés : dimension reduction, $\Gamma $-convergence, curvature and torsion, waveguides
@article{COCV_2007__13_4_793_0, author = {Bouchitt\'e, Guy and Mascarenhas, M. Lu{\'\i}sa and Trabucho, Lu{\'\i}s}, title = {On the curvature and torsion effects in one dimensional waveguides}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {793--808}, publisher = {EDP-Sciences}, volume = {13}, number = {4}, year = {2007}, doi = {10.1051/cocv:2007042}, mrnumber = {2351404}, zbl = {1139.49043}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv:2007042/} }
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 PY - 2007 SP - 793 EP - 808 VL - 13 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2007042/ DO - 10.1051/cocv:2007042 LA - en ID - COCV_2007__13_4_793_0 ER -
%0 Journal Article %A Bouchitté, Guy %A Mascarenhas, M. Luísa %A Trabucho, Luís %T On the curvature and torsion effects in one dimensional waveguides %J ESAIM: Control, Optimisation and Calculus of Variations %D 2007 %P 793-808 %V 13 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2007042/ %R 10.1051/cocv:2007042 %G en %F COCV_2007__13_4_793_0
Bouchitté, Guy; Mascarenhas, M. Luísa; Trabucho, Luís. On the curvature and torsion effects in one dimensional waveguides. ESAIM: Control, Optimisation and Calculus of Variations, Tome 13 (2007) no. 4, pp. 793-808. doi : 10.1051/cocv:2007042. http://www.numdam.org/articles/10.1051/cocv:2007042/
[1] Bloch wave homogenization and spectral asymptotic analysis. J. Math. Pures Appl. 77 (1998) 153-208. | Zbl
and ,[2] Geometrically induced discrete specrtum in curved tubes. Differ. Geometry Appl. 23 (2005) 95-105. | Zbl
, , and ,[3] Fluids and periodic structures, Research in Applied Mathematics 38. Masson, Paris (1995). | MR | Zbl
, and ,[4] An Introduction to -Convergence. Birkhäuser, Boston (1993). | MR | Zbl
,[5] Curvature-induced bounds states in quantum waveguides in two and tree dimensions. Rev. Math. Phys. 7 (1995) 73-102. | Zbl
and ,[6] Homogenization of Differential Operators and Integral Equations. Springer-Verlag, Berlin (1994). | MR
, and ,[7] On some spectral problems of mathematical physics. Partial differential equations and inverse problems., Contemp. Math. 362. Amer. Math. Soc., Providence, RI (2004) 241-276. | Zbl
,[8] Variational problems on multiply connected thin strips. II. Convergence of the Ginzburg-Landau functional. Arch. Ration. Mech. Anal. 160 (2001) 309-324. | Zbl
, ,[9] Homogenization of eigenvalue problems in perforated domains. Proc. Indian Acad. Sci. Math. Sci. 90 (1981) 239-271. | Zbl
,Cité par Sources :