Document ESAIM: COCV, Vol. 13, N°4, pp. 793-808
DOI: 10.1051/cocv:2007042
On the curvature and torsion effects in one dimensional waveguides
Guy Bouchitté1, M. Luísa Mascarenhas2 and Luís Trabucho31 Département de Mathématiques, Université du Sud-Toulon-Var, BP 132, 83957 La Garde Cedex, France; bouchitte@univ-tln.fr
2 Departamento de Matemática da F.C.T.-U.N.L. e C.M.A.-U.N.L., Quinta da Torre, 2829-516 Caparica, Portugal; mlfm@fct.unl.pt
3 Departamento de Matemática da F.C.-U.L. e C.M.A.F.-U.L., Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal; trabucho@ptmat.fc.ul.pt
(Received February 23, 2006. Revised July 7, 2006. Published online September 5, 2007.)
Abstract
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.
Mathematics Subject Classification. 49R50, 35P20, 78A50, 81Q15
Key words: Dimension reduction,
-convergence, curvature and torsion, waveguides
© EDP Sciences, SMAI 2007