On the Generalized Calderόn Formulas for Closed- and Open-Surface Elastic Scattering Problems

Abstract

The Calderόn formulas have been recently utilized in the process of constructing valid boundary integral equation systems which possess highly favorable spectral properties. This work is devoted to studying the theoretical properties of elastodynamic Calderόn formulas which provide us with a solid basis for the design of fast boundary integral equation methods solving elastic wave problems defined on a close- or open-surface in two dimensions. For the closed-surface case, it is proved that the Calderόn formula is a Fredholm operator of second-kind except for certain circumstances. For the open-surface case, we investigate weighted integral operators instead of the original integral operators which are resulted from dealing with edge singularities of potentials corresponding to the elastic scattering problems by open-surfaces, and show that the Calderόn formula is a compact perturbation of a bounded and invertible operator whose spectrum enjoys the same accumulation points as the Calderόn formula in the closed-surface case.