Inverse boundary value problems for the perturbed polyharmonic operator

TL;DR

This paper proves that for polyharmonic operators with a first order perturbation, the perturbation can be uniquely identified from boundary data in dimensions three and higher, extending inverse problem results beyond the classical case.

Contribution

It establishes the unique determination of first order perturbations of polyharmonic operators from boundary measurements, a result not valid for the standard Laplacian case.

Findings

Unique recovery of perturbations for polyharmonic operators in dimensions n≥3

Counterexamples showing non-uniqueness for the Laplacian case

Extension of inverse boundary value problem results to higher-order operators

Abstract

We show that a first order perturbation $A(x)\cdot D+q(x)$ of the polyharmonic operator $(-\Delta)^m$, $m\ge 2$, can be determined uniquely from the set of the Cauchy data for the perturbed polyharmonic operator on a bounded domain in $R^n$, $n\ge 3$. Notice that the corresponding result does not hold in general when $m=1$.