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.

Classification: 35P05, 35J05, 35B25, 35C20

Keywords: 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}, publisher = {EDP-Sciences}, volume = {17}, number = {3}, year = {2011}, pages = {887-908}, doi = {10.1051/cocv/2010028}, zbl = {1223.35248}, mrnumber = {2826984}, language = {en}, url = {http://www.numdam.org/item/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, Volume 17 (2011) no. 3, pp. 887-908. doi : 10.1051/cocv/2010028. http://www.numdam.org/item/COCV_2011__17_3_887_0/

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