Inverse boundary problems for biharmonic operators in transversally anisotropic geometries

TL;DR

This paper investigates the unique determination of first order perturbations in biharmonic operators on certain anisotropic manifolds using boundary data, relying on the injectivity of a geodesic X-ray transform.

Contribution

It establishes the uniqueness of inverse boundary problems for biharmonic operators in transversally anisotropic geometries under specific conditions.

Findings

First order perturbations are uniquely recoverable from boundary data.

Injectivity of the geodesic X-ray transform is crucial for the result.

Applicable to manifolds of dimension three or higher.

Abstract

We study inverse boundary problems for first order perturbations of the biharmonic operator on a conformally transversally anisotropic Riemannian manifold of dimension $n \ge 3$. We show that a continuous first order perturbation can be determined uniquely from the knowledge of the set of the Cauchy data on the boundary of the manifold provided that the geodesic $X$-ray transform on the transversal manifold is injective.