University of Oulu

Serov V. (2020) Some Inverse Scattering Problems for Perturbations of the Biharmonic Operator. In: Dörfler W. et al. (eds) Mathematics of Wave Phenomena. Trends in Mathematics. Birkhäuser, Cham.

Some inverse scattering problems for perturbations of the biharmonic operator

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Author: Serov, Valery1
Organizations: 1University of Oulu, Finland
Format: article
Version: accepted version
Access: embargoed
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Language: English
Published: Springer Nature, 2020
Publish Date: 2021-10-02


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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Series: Trends in mathematics
ISSN: 2297-0215
ISSN-E: 2297-024X
ISSN-L: 2297-0215
ISBN: 978-3-030-47174-3
ISBN Print: 978-3-030-47173-6
Pages: 309 - 325
DOI: 10.1007/978-3-030-47174-3_19
Host publication editor: Dörfler, Willy
Hochbruck, Marlis
Hundertmark, Dirk
Reichel, Wolfgang
Rieder, Andreas
Schnaubelt, Roland
Schörkhuber, Birgit
Conference: Conference on Mathematics of Wave Phenomena
Type of Publication: A4 Article in conference proceedings
Field of Science: 111 Mathematics
Funding: This work was supported by the Academy of Finland (application no 312123, Finnish Programme for Centres of Excellence in Inverse Modelling and Imaging 2018-2025).
Academy of Finland Grant Number: 312123
Detailed Information: 312123 (Academy of Finland Funding decision)
Copyright information: © 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