Inverse scattering and stability for the biharmonic operator

Abstract

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