On the Generalized Calderόn Formulas for Closed- and Open-Surface Elastic Scattering Problems (CSIAM Transactions on Applied Mathematics, 2022)
The Calderόn formulas have been recently utilized in the process of constructing valid boundary integral equation systems which possess highly favorablespectral properties. This work is devoted to studying the theoretical properties of elastodynamic Calderόn formulas which provide us with a solid basis for the design offast boundary integral equation methods solving elastic wave problems defined ona close- or open-surface in two dimensions. For the closed-surface case, it is provedthat the Calderόn formula is a Fredholm operator of second-kind except for certaincircumstances. For the open-surface case, we investigate weighted integral operatorsinstead of the original integral operators which are resulted from dealing with edgesingularities of potentials corresponding to the elastic scattering problems by open-surfaces, and show that the Calderόn formula is a compact perturbation of a boundedand invertible operator whose spectrum enjoys the same accumulation points as theCalderόn formula in the closed-surface case.
Keywords: Elastic wave; open-surface; Calderόn relation; Fredholm operator.