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

TL;DR

This paper demonstrates how to reconstruct a potential function in a perturbed biharmonic operator on certain manifolds using boundary measurements, extending previous uniqueness results with a constructive approach.

Contribution

It provides a constructive method to determine a potential from boundary data for the biharmonic operator on transversally anisotropic manifolds, generalizing prior uniqueness results.

Findings

Reconstruction of potential q from Dirichlet-to-Neumann map.

Applicable to 3D Euclidean domains and admissible manifolds.

Provides a constructive inversion method for the inverse boundary value problem.

Abstract

We prove that a continuous potential $q$ can be constructively determined from the knowledge of the Dirichlet-to-Neumann map for the perturbed biharmonic operator $\Delta_g^2+q$ on a conformally transversally anisotropic Riemannian manifold of dimension $\ge 3$ 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 [51]. In particular, our result is applicable and new in the case of smooth bounded domains in the $3$-dimensional Euclidean space as well as in the case of $3$-dimensional admissible manifolds.