Inverse scattering and stability for the biharmonic operator

Siamak RabieniaHaratbar

doi:10.3934/ipi.2020064

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2021, Volume 15, Issue 2: 271-283. Doi: 10.3934/ipi.2020064

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Inverse scattering and stability for the biharmonic operator

Siamak RabieniaHaratbar

Department of Mathematics, Purdue University, West Lafayette, IN 47907

Received: April 2020

Revised: August 2020

Early access: November 2020

Published: April 2021

The author is partially supported by a NSF DMS grants No. 1600327 and 1900475

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Abstract

We investigate the inverse scattering problem of the perturbed biharmonic operator by studying the recovery process of the magnetic field $ {\mathbf{A}} $ and the potential field $ V $. We show that the high-frequency asymptotic of the scattering amplitude of the biharmonic operator uniquely determines $ {\rm{curl}}\ {\mathbf{A}} $ and $ V-\frac{1}{2}\nabla\cdot{\mathbf{A}} $. We study the near-field scattering problem and show that the high-frequency asymptotic expansion up to an error $ \mathcal{O}(\lambda^{-4}) $ recovers above two quantities with no additional information about $ {\mathbf{A}} $ and $ V $. We also establish stability estimates for $ {\rm{curl}}\ {\mathbf{A}} $ and $ V-\frac{1}{2}\nabla\cdot{\mathbf{A}} $.

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Mathematics Subject Classification: 35R30, 47A40, 31B30.

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