The limit subdomain decomposition of some spectral problems in strengthened Sobolev spaces (Q5935947)

This brief paper draws attention to the use of strengthened Sobolev spaces in the context of engineering problems involving specific combinations of structures in several dimensions. Perhaps the best known example of this type, which was examined by Timoshenko in 1915 prior to the use of Hilbert spaces, was the theory of plates strengthened by rods. Other more recent examples concern three-dimensional problems in hydrodynamics with surface tension. The paper presents a number of examples in which attention is paid mainly to the behavior of segments of surfaces strengthened by rods. It is shown that the limit problems featured in the title of the paper are problems that arise when homogeneous Dirichlet boundary conditions occur on a surface which is partitioned into several parts. Estimates are given for the closeness of eigenvalues and these are shown to be optimal. Generalizations of problems in elasticity, hydrodynamics and the theory of shallow shells strengthened by ribs are also presented.