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.

We consider the Laplace operator in a thin tube of ${ℝ}^{3}$ 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 $\Gamma$-convergence theorem for a suitable sequence of quadratic energies.

DOI : https://doi.org/10.1051/cocv:2007042
Classification : 49R50,  35P20,  78A50,  81Q15
Mots clés : dimension reduction, $\Gamma$-convergence, curvature and torsion, waveguides
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author = {Bouchitte, 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},
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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/

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