Singularities of solutions to the boundary value problems for elastic and Maxwell’s equations. (English) Zbl 0669.73017

Author has studied the propagation of singularities of solutions of the boundary value problems for elastic equations with free boundary conditions in an isotropic medium and Maxwell’s equations in a vacuum region by a perfect conductor. In the case of elastic equations, all the points of contangential space of the boundary are classified into five classes. The propagation of singularities arising from a point in each class has been analyzed. The author has shown that the behaviour of the singularities of the solution for Maxwell’s equation is similar to that of the solution for the Dirichlet problem. As application of the theorems on the propagation of singularities, it has been demonstrated that for non-trapping obstacles the decay of locel energy is exponential. For a convex obstacle it is possible to check all the points of singular support of the scattering kernel.
The paper is divided into two chapters. The first one deals with the elastic wave equation and the second one deals with Maxwell’s equation. In the Appendix, theorems on propagation of singularities of the solutions to boundary value problems for higher order single strictly hyperbolic equations and first order hyperbolic systems with generalized Agmon-Lopatinski conditions are given.
Reviewer: K.N.Srivastava


74J20 Wave scattering in solid mechanics
35L67 Shocks and singularities for hyperbolic equations
35R35 Free boundary problems for PDEs