TY - JOUR
T1 - On the Generalized Calderόn Formulas for Closed- and Open-Surface Elastic Scattering Problems
AU - Xu , Liwei
AU - Yin , Tao
JO - CSIAM Transactions on Applied Mathematics
VL - 4
SP - 601
EP - 625
PY - 2022
DA - 2022/11
SN - 3
DO - http://doi.org/10.4208/csiam-am.SO-2021-0050
UR - https://global-sci.org/intro/article_detail/csiam-am/21150.html
KW - Elastic wave, open-surface, Calderόn relation, Fredholm operator.
AB - <p style="text-align: justify;">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.</p>