Boundary stabilization of elastodynamic systems. I: Rellich-type relations for a problem in elasticity involving singularities
This work is concerned with mixed boundary value problems in homogeneous and isotropic linear elastostatics whose field equations are known as Lamé's equations. The domain involved is a bounded open set in a Euclidean space. The boundary of the domain is assumed to be the union of two sets in one of which displacements are prescribed (Dirichlet conditions) while in the other tractions are given (Neumann conditions). It is supposed that the boundary data belong to appropriate Sobolev spaces. The nteriors of these sets are disjoint whereas their closure may intersect at the interface. If the interface is not empty, i.e., if the boundary is a connected set, then it is known that singularities occur in the solution. In this work Rellich type integral relations are obtained for the following cases: (i) \(n\)-dimensional domain with no interface, (ii) bidimensional polygonal domain with two interface points, (iii) smooth bidimensional domain with two interface points, (iv) smooth \(n\)-dimensional domain with an interface. The authors state that the relations obtained will be useful in the problem of boundary stabilization. But this will be the subject of a future work.
scientific article; zbMATH DE number 5712197