Recovering nonsmooth coefficients for higher-order perturbations of a polyharmonic operator

TL;DR

This paper investigates the inverse problem of recovering nonsmooth coefficients in higher-order polyharmonic operators, establishing injectivity of the coefficient-to-bilinear form map under low regularity conditions.

Contribution

It demonstrates the unique recoverability of coefficients from boundary data for polyharmonic operators with minimal regularity assumptions.

Findings

Injectivity of the coefficient-to-bilinear form map.

Results applicable under low regularity of coefficients.

Equivalence to boundary measurements like Dirichlet to Neumann map.

Abstract

We consider an inverse problem for a higher order elliptic operator where the principal part is the polyharmonic operator $(-\Delta)^m$ with $ m \geq 2$. We show that the map from the coefficients to a certain bilinear form is injective. We have a particular focus on obtaining these results under lower regularity on the coefficients. It is known that knowledge of this bilinear form is equivalent to a knowledge of a Dirichlet to Neumann map or the Cauchy data for solutions.