Reconstructing a potential perturbation of the biharmonic operator on transversally anisotropic manifolds

Abstract

<p style='text-indent:20px;'>We prove that a continuous potential <inline-formula><tex-math id="M1">\begin{document}$ q $\end{document}</tex-math></inline-formula> can be constructively determined from the knowledge of the Dirichlet–to–Neumann map for the perturbed biharmonic operator <inline-formula><tex-math id="M2">\begin{document}$ \Delta_g^2+q $\end{document}</tex-math></inline-formula> on a conformally transversally anisotropic Riemannian manifold of dimension <inline-formula><tex-math id="M3">\begin{document}$ \ge 3 $\end{document}</tex-math></inline-formula> with boundary, assuming that the geodesic ray transform on the transversal manifold is constructively invertible. This is a constructive counterpart of the uniqueness result of [<xref ref-type="bibr" rid="b56">56</xref>]. In particular, our result is applicable and new in the case of smooth bounded domains in the <inline-formula><tex-math id="M4">\begin{document}$ 3 $\end{document}</tex-math></inline-formula>–dimensional Euclidean space as well as in the case of <inline-formula><tex-math id="M5">\begin{document}$ 3 $\end{document}</tex-math></inline-formula>–dimensional admissible manifolds.</p>