English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

The Calder\'on formulas (i.e., the combination of single-layer andhyper-singular boundary integral operators) have been widely utilized in theprocess of constructing valid boundary integral equation systems which couldpossess highly favorable spectral properties. This work is devoted to studyingthe theoretical properties of elastodynamic Calder\'on formulas which provideus with a solid basis for the design of fast boundary integral equation methodssolving elastic wave problems defined on a close-surface or an open-surface intwo dimensions. For the closed-surface case, it is proved that the Calder\'onformula is a Fredholm operator of second-kind except for certain circumstances.Regarding to the open-surface case, we investigate weighted integral operatorsinstead of the original integral operators which are resulted from dealing withedge singularities of potentials corresponding to the elastic scatteringproblems by open-surfaces, and show that the Calder\'on formula is a compactperturbation of a bounded and invertible operator. To complete the proof, weneed to use the well-posedness result of the elastic scattering problem, theanalysis of the zero-frequency integral operators defined on the straight arc,the singularity decompositions of the kernels of integral operators, and a newrepresentation formula of the hyper-singular operator. Moreover, it can bedemonstrated that the accumulation point of the spectrum of the invertibleoperator is the same as that of the eigenvalues of the Calder\'on formula inthe closed-surface case.

On the generalized Calder\'on formulas for closed- and open-surface elastic scattering problems