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.

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.

DOI : 10.1051/cocv/2010028
Classification : 35P05, 35J05, 35B25, 35C20
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] N.S. Bakhvalov and G.P. Panasenko, Homogenization: Averaging processes in periodic media. Kluwer, Dordrecht/Boston/ London (1989). | Zbl

[2] D. Borisov and P. Freitas, 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

[3] G. Bouchitté, M.L. Mascarenhas and L. Trabucho, On the curvature and torsion effects in one dimensional waveguides. ESAIM: COCV 13 (2007) 793-808. | Numdam | MR | Zbl

[4] G. Cardone, T. Durante and S.A. Nazarov, 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

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

[6] P. Freitas and D. Krejčiřík, Location of the nodal set for thin curved tubes. Indiana Univ. Math. J. 57 (2008) 343-376. | MR | Zbl

[7] L. Friedlander and M. Solomyak, 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

[8] L. Friedlander and M. Solomyak, On the spectrum of the Dirichlet Laplacian in a narrow strip. Israel J. Math. 170 (2009) 337-354. | MR | Zbl

[9] D. Grieser, 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] D. Krejčiřík, 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] V.P. Mikhajlov, Partial differential equations. Moscow, Mir Publishers (1978). | MR | Zbl

[12] S.A. Nazarov, Dimension Reduction and Integral Estimates, Asymptotic Theory of Thin Plates and Rods 1. Novosibirsk, Nauchnaya Kniga (2001).

[13] O.A. Oleinik, A.S. Shamaev and G.A. Yosifyan, Mathematical problems in elasticity and homogenization, Studies in Mathematics and its Applications 26. Amsterdam etc., North-Holland (1992). | MR | Zbl

[14] G.P. Panasenko and M.E. Perez, Asymptotic partial decomposition of domain for spectral problems in rod structures. J. Math. Pures Appl. 87 (2007) 1-36. | MR | Zbl

[15] M.I. Vishik and L.A. Lyusternik, 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

Cité par Sources :