Determination of Lower Order Perturbations of a Polyharmonic Operator in Two Dimensions

TL;DR

This paper investigates an inverse boundary value problem for polyharmonic operators in two dimensions, demonstrating that Cauchy data can uniquely identify certain anisotropic perturbations of the operator.

Contribution

It introduces a novel approach combining ar{ ext{}}}-techniques and stationary phase methods to establish uniqueness results for perturbations of polyharmonic operators.

Findings

Unique determination of anisotropic perturbations of orders up to m-1.

Partial uniqueness for higher order perturbations under specific restrictions.

Application of ar{ ext{}}}-techniques in inverse boundary problems.

Abstract

We study an inverse boundary value problem for a polyharmonic operator in two dimensions. We show that the Cauchy data uniquely determine all the anisotropic perturbations of orders at most $m-1$ and several perturbations of orders $m$ to $2m-2$ under some restriction. The uniqueness proof relies on the $\bar{\partial}$-techniques and the method of stationary phase.