On the structure of the spectrum for the elasticity problem in a body with a supersharp spike. (Russian, English) Zbl 1224.35395

Bakharev, F. L.; Nazarov, S. A.

Sib. Mat. Zh. 50, No. 4, 746-756 (2009); translation in Sib. Math. J. 50, No. 4, 587-595 (2009).

Summary: We establish that the continuous spectrum of the Neumann problem for the system of elasticity equations occupies the entire closed positive real semiaxis in the case that a three-dimensional body with a sharp-spiked cusp whose cross-section contracts to a point with the velocity \(O(r^{1+\gamma})\), where \(r\) is the distance to the vertex of the spike and \(\gamma > 1\) is the sharpness exponent.

MSC:

35Q74	PDEs in connection with mechanics of deformable solids
35P05	General topics in linear spectral theory for PDEs
74B05	Classical linear elasticity

Keywords:

system of elasticity equations; spike; cusp; peak; continuous spectrum