Eigenvalue asymptotics of long Kirchhoff plates with clamped edges

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.