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Journal of Spectral Theory


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Volume 6, Issue 1, 2016, pp. 145–183
DOI: 10.4171/JST/122

Published online: 2016-04-04

Inverse boundary problems for polyharmonic operators with unbounded potentials

Katsiaryna Krupchyk[1] and Gunther Uhlmann[2]

(1) University of California at Irvine, USA
(2) University of Washington, Seattle, United States

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.

Keywords: Inverse boundary problem, polyharmonic operator, unbounded potential, Carleman estimate, Green function

Krupchyk Katsiaryna, Uhlmann Gunther: Inverse boundary problems for polyharmonic operators with unbounded potentials. J. Spectr. Theory 6 (2016), 145-183. doi: 10.4171/JST/122