University of Oulu

Teemu Tyni, Valery Serov. Scattering problems for perturbations of the multidimensional biharmonic operator. Inverse Problems & Imaging, 2018, 12 (1) : 205-227. doi: 10.3934/ipi.2018008

Scattering problems for perturbations of the multidimensional biharmonic operator

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Author: Tyni, Teemu1; Serov, Valery1
Organizations: 1Department of Mathematical Sciences, FI-90014 University of Oulu, Finland
Format: article
Version: accepted version
Access: open
Online Access: PDF Full Text (PDF, 0.4 MB)
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Language: English
Published: American Institute of Mathematical Sciences, 2018
Publish Date: 2019-04-04


Some scattering problems for the multidimensional biharmonic operator are studied. The operator is perturbed by first and zero order perturbations, which maybe complex-valued and singular. We show that the solutions to direct scattering problem satisfy a Lippmann-Schwinger equation, and that this integral equation has a unique solution in the weighted Sobolev space \(H_{-δ}^2\). The main result of this paper is the proof of Saito’s formula, which can be used to prove a uniqueness theorem for the inverse scattering problem. The proof of Saito’s formula is based on norm estimates for the resolvent of the direct operator in \(H_{-δ}^1\).

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Series: Inverse problems and imaging
ISSN: 1930-8337
ISSN-E: 1930-8345
ISSN-L: 1930-8337
Volume: 12
Issue: 1
Pages: 205 - 227
DOI: 10.3934/ipi.2018008
Type of Publication: A1 Journal article – refereed
Field of Science: 111 Mathematics
Funding: This work was supported by the Academy of Finland (application number 250215, Finnish Programme for Centres of Excellence in Research 2012–2017). The first author was supported by the Doctoral Programme of Exact Sciences at the University of Oulu, Finland.
Copyright information: Inverse Problems & Imaging © 2018 Published by AIMS. All rights reserved.