Zur Lösung räumlicher Probleme der Elastizitätstheorie für torusberandete Gebiete mit Potentialen vom Typ der einfachen Schicht

Abstract

The problem of the elastic homogeneous and isotropic whole space, with a simple connected inclusion of other material in clastotatics and for stationary elastic vibrations is considered. On the contact surface $L$ the jumps of displacements and tractions are given. $L$ is assumed to be homeomorphic to the boundary surface of a regular torus, and to belong to the class $C^{1,\beta }$. Using the elastic single layer potential, a system of integral equations is obtained, which is transformed to a twodimensional singular integral equation system by the aid of a Beltrami differential operator $\Omega$. The singular system is normal solvable and has the index 0. Studying the kernel and the range of the operator $\Omega$, the existence of solutions of the original problems can be proved. Some generalisations of the approach on other problems are discussed.