Complete asymptotic expansions for eigenvalues of Dirichlet Laplacian in thin three-dimensional rods*
Denis Borisov; Giuseppe Cardone
ESAIM: Control, Optimisation and Calculus of Variations (2011)
- Volume: 17, Issue: 3, page 887-908
- ISSN: 1292-8119
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topBorisov, Denis, and Cardone, Giuseppe. "Complete asymptotic expansions for eigenvalues of Dirichlet Laplacian in thin three-dimensional rods*." ESAIM: Control, Optimisation and Calculus of Variations 17.3 (2011): 887-908. <http://eudml.org/doc/221911>.
@article{Borisov2011,
abstract = {
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.
},
author = {Borisov, Denis, Cardone, Giuseppe},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Thin rod; Dirichlet Laplacian; eigenvalue; asymptotics; thin rod; asymptotic expansions},
language = {eng},
month = {8},
number = {3},
pages = {887-908},
publisher = {EDP Sciences},
title = {Complete asymptotic expansions for eigenvalues of Dirichlet Laplacian in thin three-dimensional rods*},
url = {http://eudml.org/doc/221911},
volume = {17},
year = {2011},
}
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
DA - 2011/8//
PB - EDP Sciences
VL - 17
IS - 3
SP - 887
EP - 908
AB -
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.
LA - eng
KW - Thin rod; Dirichlet Laplacian; eigenvalue; asymptotics; thin rod; asymptotic expansions
UR - http://eudml.org/doc/221911
ER -
References
top- N.S. Bakhvalov and G.P. Panasenko, Homogenization: Averaging processes in periodic media. Kluwer, Dordrecht/Boston/ London (1989).
- D. Borisov and P. Freitas, Singular asymptotic expansions for Dirichlet eigenvalues and eigenfunctions on thin planar domains. Ann. Inst. Henri Poincaré, Anal. Non Linéaire26 (2009) 547–560.
- G. Bouchitté, M.L. Mascarenhas and L. Trabucho, On the curvature and torsion effects in one dimensional waveguides. ESAIM: COCV13 (2007) 793–808.
- 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).
- P. Duclos and P. Exner, Curvature-induced bound states in quantum waveguides in two and three dimensions. Rev. Math. Phys.7 (1995) 73–102.
- P. Freitas and D. Krejčiřík, Location of the nodal set for thin curved tubes. Indiana Univ. Math. J.57 (2008) 343–376.
- 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).
- L. Friedlander and M. Solomyak, On the spectrum of the Dirichlet Laplacian in a narrow strip. Israel J. Math.170 (2009) 337–354.
- 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).
- D. Krejčiřík, Spectrum of the Laplacian in a narrow curved strip with combined Dirichlet and Neumann boundary conditions. ESAIM: COCV15 (2009) 555–568.
- V.P. Mikhajlov, Partial differential equations. Moscow, Mir Publishers (1978).
- S.A. Nazarov, Dimension Reduction and Integral Estimates, Asymptotic Theory of Thin Plates and Rods1. Novosibirsk, Nauchnaya Kniga (2001).
- O.A. Oleinik, A.S. Shamaev and G.A. Yosifyan, Mathematical problems in elasticity and homogenization, Studies in Mathematics and its Applications26. Amsterdam etc., North-Holland (1992).
- 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.
- 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.
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