Inverse boundary value problem of determining up to second order tensors appear in the lower order perturbations of the polyharmonic operator

TL;DR

This paper proves that the coefficients of a perturbed polyharmonic operator, including a second order tensor, can be uniquely recovered from boundary measurements, advancing inverse boundary value problem theory.

Contribution

It demonstrates the unique determination of second order tensor coefficients in a polyharmonic operator from boundary data, extending inverse boundary value problem results.

Findings

Coefficients A, B, q are recoverable from Dirichlet-to-Neumann data.

The result includes the determination of a second order symmetric tensor.

Provides a new example of inverse boundary problem for higher order tensors.

Abstract

We consider the following perturbed polyharmonic operator $\Lc(x,D)$ of order $2m$ defined in a bounded domain $\Omega \subset \mathbb{R}^n, n\geq 3$ with smooth boundary, as \begin{equation*} \Lc(x,D) \equiv (-\Delta)^m + \sum_{j,k=1}^{n}A_{jk} D_{j}D_{k} + \sum_{j=1}^{n}B_{j} D_{j} + q(x), \end{equation*} where $A$ is a symmetric $2$-tensor field, $B$ and $q$ are vector field and scalar potential respectively. We show that the coefficients $A=[A_{jk}]$, $B=(B_j)$ and $q$ can be recovered from the associated Dirichlet-to-Neumann data on the boundary. Note that, this result shows an example of determining higher order ($2$nd order) symmetric tensor field in the class of inverse boundary value problem.