We consider the Dirichlet Laplacian in a thin curved three-dimensional rod. The rod is finite. Its cross-section is constant and small, and rotates along the reference curve in an arbitrary way. We find a two-parametric set of the eigenvalues of such operator and construct their complete asymptotic expansions. We show that this two-parametric set contains any prescribed number of the first eigenvalues of the considered operator. We obtain the complete asymptotic expansions for the eigenfunctions associated with these first eigenvalues.
Mots clés : thin rod, Dirichlet laplacian, eigenvalue, asymptotics
@article{COCV_2011__17_3_887_0, author = {Borisov, Denis and Cardone, Giuseppe}, title = {Complete asymptotic expansions for eigenvalues of {Dirichlet} laplacian in thin three-dimensional rods}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {887--908}, publisher = {EDP-Sciences}, volume = {17}, number = {3}, year = {2011}, doi = {10.1051/cocv/2010028}, mrnumber = {2826984}, zbl = {1223.35248}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv/2010028/} }
TY - JOUR AU - Borisov, Denis AU - Cardone, Giuseppe TI - Complete asymptotic expansions for eigenvalues of Dirichlet laplacian in thin three-dimensional rods JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2011 SP - 887 EP - 908 VL - 17 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2010028/ DO - 10.1051/cocv/2010028 LA - en ID - COCV_2011__17_3_887_0 ER -
%0 Journal Article %A Borisov, Denis %A Cardone, Giuseppe %T Complete asymptotic expansions for eigenvalues of Dirichlet laplacian in thin three-dimensional rods %J ESAIM: Control, Optimisation and Calculus of Variations %D 2011 %P 887-908 %V 17 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2010028/ %R 10.1051/cocv/2010028 %G en %F COCV_2011__17_3_887_0
Borisov, Denis; Cardone, Giuseppe. Complete asymptotic expansions for eigenvalues of Dirichlet laplacian in thin three-dimensional rods. ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 3, pp. 887-908. doi : 10.1051/cocv/2010028. http://www.numdam.org/articles/10.1051/cocv/2010028/
[1] Homogenization: Averaging processes in periodic media. Kluwer, Dordrecht/Boston/ London (1989). | Zbl
and ,[2] Singular asymptotic expansions for Dirichlet eigenvalues and eigenfunctions on thin planar domains. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 26 (2009) 547-560. | Numdam | MR | Zbl
and ,[3] On the curvature and torsion effects in one dimensional waveguides. ESAIM: COCV 13 (2007) 793-808. | Numdam | MR | Zbl
, and ,[4] The localization effect for eigenfunctions of the mixed boundary value problem in a thin cylinder with distorted ends. SIAM J. Math. Anal. (to appear). | MR | Zbl
, and ,[5] Curvature-induced bound states in quantum waveguides in two and three dimensions. Rev. Math. Phys. 7 (1995) 73-102. | MR | Zbl
and ,[6] Location of the nodal set for thin curved tubes. Indiana Univ. Math. J. 57 (2008) 343-376. | MR | Zbl
and ,[7] On the spectrum of the Dirichlet Laplacian in a narrow infinite strip, in Spectral theory of differential operators: M. Sh. Birman 80th anniversary collection, Adv. Math. Sci. 225, T. Suslina and D. Yafaev Eds., AMS Translations - Series 2, Providence (2008). | MR | Zbl
and ,[8] On the spectrum of the Dirichlet Laplacian in a narrow strip. Israel J. Math. 170 (2009) 337-354. | MR | Zbl
and ,[9] Thin tubes in mathematical physics, global analysis and spectral geometry, in Analysis on Graphs and Its Applications, P. Exner, J.P. Keating, P. Kuchment, T. Sunada and A. Teplyaev Eds., Proc. Symp. Pure Math. 77, AMS, Providence (2008). | MR | Zbl
,[10] Spectrum of the Laplacian in a narrow curved strip with combined Dirichlet and Neumann boundary conditions. ESAIM: COCV 15 (2009) 555-568. | Numdam | MR | Zbl
,[11] Partial differential equations. Moscow, Mir Publishers (1978). | MR | Zbl
,[12] Dimension Reduction and Integral Estimates, Asymptotic Theory of Thin Plates and Rods 1. Novosibirsk, Nauchnaya Kniga (2001).
,[13] Mathematical problems in elasticity and homogenization, Studies in Mathematics and its Applications 26. Amsterdam etc., North-Holland (1992). | MR | Zbl
, and ,[14] Asymptotic partial decomposition of domain for spectral problems in rod structures. J. Math. Pures Appl. 87 (2007) 1-36. | MR | Zbl
and ,[15] The asymptotic behaviour of solutions of linear differential equations with large or quickly changing coefficients and boundary conditions. Russ. Math. Surv. 15 (1960) 23-91. | MR | Zbl
and ,Cité par Sources :