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 ℝ3 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.

DOI : 10.1051/cocv:2007042
Classification : 49R50, 35P20, 78A50, 81Q15
Mots-clés : dimension reduction, Γ-convergence, curvature and torsion, waveguides 
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     title = {On the curvature and torsion effects in one dimensional waveguides},
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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. https://numdam.org/articles/10.1051/cocv:2007042/

[1] G. Allaire and C. Conca, Bloch wave homogenization and spectral asymptotic analysis. J. Math. Pures Appl. 77 (1998) 153-208. | Zbl

[2] B. Chenaud, P. Duclos, P. Freitas and D. Krejčiřík, Geometrically induced discrete specrtum in curved tubes. Differ. Geometry Appl. 23 (2005) 95-105. | Zbl

[3] C. Conca, J. Planchard and M. Vanninathan, Fluids and periodic structures, Research in Applied Mathematics 38. Masson, Paris (1995). | MR | Zbl

[4] G. Dal Maso, An Introduction to Γ-Convergence. Birkhäuser, Boston (1993). | MR | Zbl

[5] P. Duclos and P. Exner, Curvature-induced bounds states in quantum waveguides in two and tree dimensions. Rev. Math. Phys. 7 (1995) 73-102. | Zbl

[6] V. Jikov, S.M. Kozlov and O.A. Oleinik, Homogenization of Differential Operators and Integral Equations. Springer-Verlag, Berlin (1994). | MR

[7] 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. | Zbl

[8] 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. | Zbl

[9] M. Vanninathan, Homogenization of eigenvalue problems in perforated domains. Proc. Indian Acad. Sci. Math. Sci. 90 (1981) 239-271. | Zbl

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