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Complete asymptotic expansions for eigenvalues ofDirichlet 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
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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.

Keywords

Thin rodDirichlet Laplacianeigenvalueasymptotics
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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