Unique determination of anisotropic perturbations of a polyharmonic operator from partial boundary data

TL;DR

This paper establishes the unique recovery of anisotropic tensor perturbations in polyharmonic operators from partial boundary data, introducing a new approach using generalized momentum ray transforms and CGO solutions.

Contribution

It introduces the first inversion of generalized momentum ray transforms for symmetric tensor fields in Calderón-type inverse problems involving polyharmonic operators.

Findings

Proves uniqueness of tensor perturbations from partial boundary data.

Develops a new inversion formula for generalized momentum ray transforms.

Constructs CGO solutions for higher order polyharmonic operators.

Abstract

We study an inverse problem involving the unique recovery of several lower order anisotropic tensor perturbations of a polyharmonic operator in a bounded domain from the knowledge of the Dirichlet to Neumann map on a part of boundary. The uniqueness proof relies on the inversion of generalized momentum ray transforms (MRT) for symmetric tensor fields, which we introduce for the first time to study Calder\'on-type inverse problems. We construct suitable complex geometric optics (CGO) solutions for the polyharmonic operators that reduces the inverse problem to uniqueness results for a generalized MRT. The uniqueness result and the inversion formula we prove for generalized MRT could be of independent interest and we expect it to be applicable to other inverse problems for higher order operators involving tensor perturbations.