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Complete asymptotic expansions for eigenvaluesofDirichlet Laplacian in thin three-dimensional rods*

Published online by Cambridge University Press:  06 August 2010

Denis Borisov and
Giuseppe Cardone
Denis Borisov
Affiliation:
Bashkir State Pedagogical University, October Revolution St. 3a, 450000 Ufa, Russia. borisovdi@yandex.ru
Giuseppe Cardone
Affiliation:
University of Sannio, Department of Engineering, Corso Garibaldi, 107, 82100 Benevento, Italy. gcardone@unisannio.it

Abstract

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We consider the Dirichlet Laplacian in a thin curvedthree-dimensional rod. The rod is finite. Its cross-section isconstant and small, and rotates along the reference curve in anarbitrary way. We find a two-parametric set of the eigenvalues ofsuch operator and construct their complete asymptotic expansions. Weshow that this two-parametric set contains any prescribed number ofthe first eigenvalues of the considered operator. We obtain thecomplete asymptotic expansions for the eigenfunctions associatedwith these first eigenvalues.

Keywords

Thin rodDirichlet Laplacianeigenvalueasymptotics

Information

Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 17 , Issue 3 , July 2011, pp. 887 - 908
Copyright
© EDP Sciences, SMAI, 2010

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