Inverse problems for the perturbed polyharmonic operator with coefficients in Sobolev spaces with non-positive order

Abstract

We show that the knowledge of the Dirichlet-to-Neumann map on the boundary of a bounded open set in $\mathbb R^n$, $n\ge 3$, for the perturbed polyharmonic operator $(-\Delta)^m+A\cdot D+q$, $m\ge 2$, with $n>m$, $A\in W^{-\frac{m-2}{2},\frac{2n}{m}}$ and $q\in W^{-\frac{m}{2}+\delta,\frac{2n}{m}}$, with $0<\delta<1/2$, determines the potentials $A$ and $q$ in the set uniquely. The proof is based on a Carleman estimate with linear weights and with a gain of two derivatives and on the property of products of functions in Sobolev spaces.