Inverse boundary problems for polyharmonic operators with unbounded potentials

TL;DR

This paper proves the unique determination of unbounded potentials in polyharmonic operators from boundary measurements, using advanced Green function constructions and Carleman estimates.

Contribution

It introduces a method to recover unbounded potentials in polyharmonic operators from boundary data, expanding inverse boundary problem theory.

Findings

Unique determination of potential q from Dirichlet-to-Neumann map

Construction of a special Green function with specific mapping properties

Derivation of L^p estimates using Carleman estimates

Abstract

We show that the knowledge of the Dirichlet-to-Neumann map on the boundary of a bounded open set in $R^n$ for the perturbed polyharmonic operator $(-\Delta)^m +q$ with $q\in L^{n/2m}$, $n>2m$, determines the potential $q$ in the set uniquely. In the course of the proof, we construct a special Green function for the polyharmonic operator and establish its mapping properties in suitable weighted $L^2$ and $L^p$ spaces. The $L^p$ estimates for the special Green function are derived from $L^p$ Carleman estimates with linear weights for the polyharmonic operator.