Boundary integral equations of elasticity in domains with piecewise smooth boundaries

Abstract

In the author's papers [1-~ a method for investigation of boundary integral equations arising in problems of mechanics of continuum in domains with pieeewise smooth boundaries was proposed. Traditionally, equations of the potential theory are studied directly by methods of the Fredholm and singular integral operator theories. In the case of a nonsmooth boundary this way leads to difficulties that have not been overcome until now. In a sense our approach is opposite to the traditional one. It is based on the well-known fact that solutions of integral equations can be expressed in terms of solutions of some exterior and interior boundary value problems. These are studied with the help of the theory developed in [4 8] and, as a result, theorems on solvability of equations of the potential theory are obtained. For these equations we can get differentiability properties and asymptotics of solutions near singularities of the boundary using the same approach ~, 9 l j. In [~ our method of construction of the potential theory was illustrated by the example of three boundary value problems of linear isotropic elasticity, namely, the first, the second and mixed, as well as of the same problems for the Laplace operator under the hypothesis that there exist a finite number of non-intersecting smooth edges on the boundary. In the present lecture we study the first two boundary value problems for the Lam~ system in domains with boundary singularities of the type of edges, conic points and polyhedral angles. New results on the harmonic potential theory are also reported.