Some inverse scattering problems for perturbations of the biharmonic operator
Serov, Valery (2020-10-02)
Serov V. (2020) Some Inverse Scattering Problems for Perturbations of the Biharmonic Operator. In: Dörfler W. et al. (eds) Mathematics of Wave Phenomena (pp. 309-325). Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-47174-3_19
© Springer Nature Switzerland AG 2020. This is a post-peer-review, pre-copyedit version of an article published in Trends in Mathematics. The final authenticated version is available online at https://doi.org/10.1007/978-3-030-47174-3_19.
https://rightsstatements.org/vocab/InC/1.0/
https://urn.fi/URN:NBN:fi-fe2020110989747
Tiivistelmä
Abstract
Some inverse scattering problems for the three-dimensional biharmonic operator are considered. The operator is perturbed by first and zero order perturbations, which may be complex-valued and singular. We show the existence of the scattering solutions in the Sobolev space \(W^1_{\infty }(R^3)\). One of the main result of this paper is the proof of analogue of Saito’s formula (in different form as known before), which can be used to prove a uniqueness theorem for the inverse scattering problem. Another main result is to obtain the estimates for the kernel of the resolvent of the direct operator in \(W^1_{\infty}\) and to prove the reconstruction formula for the unknown coefficients of this perturbation.
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