Inverse boundary problems for polyharmonic operators with unbounded potentials

Abstract

We show that the knowledge of the Dirichlet–to–Neumann map on the boundary of a bounded open set in ${\mathbb{R}}^n$ for the perturbed polyharmonic operator $(-\Delta {)}^m+q$ with $q\in L^{\frac{n}{2m}}$, $n>2m$, 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.