Determining the first order perturbation of a polyharmonic operator on admissible manifolds

TL;DR

This paper proves unique determination of first order perturbations in a polyharmonic operator on certain manifolds using boundary measurements, extending inverse problem techniques with complex geometrical optics solutions.

Contribution

It establishes the first uniqueness result for the inverse boundary value problem for a first order perturbation of a polyharmonic operator on admissible manifolds.

Findings

Dirichlet-to-Neumann map determines $X$ and $q$ uniquely

Construction of complex geometrical optics solutions using Carleman estimates

Uniqueness does not hold for the Laplace-Beltrami operator perturbation

Abstract

We consider the inverse boundary value problem for the first order perturbation of the polyharmonic operator $\mathcal L_{g,X,q}$, with $X$ being a $W^{1,\infty}$ vector field and $q$ being an $L^\infty$ function on compact Riemannian manifolds with boundary which are conformally embedded in a product of the Euclidean line and a simple manifold. We show that the knowledge of the Dirichlet-to-Neumann determines $X$ and $q$ uniquely. The method is based on the construction of complex geometrical optics solutions using the Carleman estimate for the Laplace-Beltrami operator due to Dos Santos Ferreira, Kenig, Salo and Uhlmann. Notice that the corresponding uniqueness result does not hold for the first order perturbation of the Laplace-Beltrami operator.