Complete asymptotic expansions for eigenvalues of Dirichlet Laplacian in thin three-dimensional rods*
Bashkir State Pedagogical University, October
Revolution St. 3a, 450000 Ufa, Russia. firstname.lastname@example.org
2 University of Sannio, Department of Engineering, Corso Garibaldi, 107, 82100 Benevento, Italy. email@example.com
Revised: 22 February 2010
Revised: 6 April 2010
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.
Mathematics Subject Classification: 35P05 / 35J05 / 35B25 / 35C20
Key words: Thin rod / Dirichlet Laplacian / eigenvalue / asymptotics
D.B. was partially supported by RFBR (10-01-00118), by the grant of the President of Russia for young scientists-doctors of sciences (MD-453.2010.1) and for Leading Scientific School (NSh-6249.2010.1), by Federal Task Program (contract 02.740.11. 0612), and by the grant of FCT (ptdc/mat/101007/2008). The work was partially supported by the project “Progetto ISA: Attivita’ di Internazionalizzazione dell Universita’ degli Studi del Sannio”.
© EDP Sciences, SMAI, 2010