Eigenvalue asymptotics of long Kirchhoff plates with clamped edges

F. L. Bakharev ; S. A. Nazarov

Sbornik. Mathematics, Tome 210 (2019) no. 4, pp. 473-494

Résumé

Asymptotic expansions are constructed for the eigenvalues and eigenfunctions of the Dirichlet problem for the biharmonic operator in thin domains (Kirchhoff plates with clamped edges). For a rectangular plate the leading terms are asymptotically determined from the Dirichlet problem for a second-order ordinary differential equation, while for a $\mathsf T$-junction of plates they are determined from another limiting problem in an infinite waveguide formed by three half-strips in the shape of a letter $\mathsf T$ and describing a boundary-layer phenomenon. Open questions are stated for which the method developed gives no answer. Bibliography: 33 titles.

Détail

Keywords: Kirchhoff plate, eigenvalues and eigenfunctions, asymptotic behaviour, dimension reduction, boundary layer.

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