The spectrum of the elasticity problem for a spiked body. (Russian, English) Zbl 1224.35397

Nazarov, S. A.

Sib. Mat. Zh. 49, No. 5, 1105-1127 (2008); translation in Sib. Math. J. 49, No. 5, 874-893 (2008).

Summary: We establish the existence of continuous spectrum for the operator of the linear elasticity problem in a three-dimensional domain with a sufficiently sharp spiked singularity of the boundary. We obtain some information about the structure of the spectrum and verify the weighted Korn inequality, which enables us to prove that the spectrum is discrete for insufficiently sharp spikes. We state some open questions.

MSC:

35Q74	PDEs in connection with mechanics of deformable solids
74B05	Classical linear elasticity

Keywords:

elasticity equations; zero cusp; spike; discrete spectrum; continuous spectrum