ESAIM: Control, Optimisation and Calculus of Variations

Table of Contents - October 2007 - Volume 13, Issue 04  

Research Article

On the curvature and torsion effects in one dimensional waveguides

Bouchitté, Guya1, Mascarenhas, M. Luísaa2 and Trabucho, Luísa3

a1 Département de Mathématiques, Université du Sud-Toulon-Var, BP 132, 83957 La Garde Cedex, France; bouchitte@univ-tln.fr

a2 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

a3 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

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.

(Received February 23 2006)

(Revised July 7 2006)

(Online publication September 5 2007)

Key Words:

  • Dimension reduction;
  • Γ-convergence;
  • curvature and torsion;
  • waveguides

Mathematics Subject Classification:

  • 49R50;
  • 35P20;
  • 78A50;
  • 81Q15