On Invertibility of Elastic Single-Layer Potential Operator

Abstract

The question of unique solvability of the boundary integral equation of the first kind given by the single-layer potential operator is studied in the case of plane isotropic elasticity. First, a sufficient condition of the positivity, and hence invertibility, of this operator is presented. Then, considering a scale transformation of the domain boundary, the well known formula for scaling the Robin constant in potential theory is generalized to elasticity. Subsequently, an explicit equation for evaluation of critical scales for a given boundary, when the single-layer operator fails to be invertible, is deduced. It is proved that there are either two simple critical scales or one double critical scale for any domain boundary. Numerical results, obtained applying a symmetric Galerkin boundary element code, confirm the propositions of the theory developed for both single and multi-contour boundaries.