Inverse boundary problems for polyharmonic operators with unbounded potentials

  • Katsiaryna Krupchyk

    University of California at Irvine, USA
  • Gunther Uhlmann

    University of Washington, Seattle, United States

Abstract

We show that the knowledge of the Dirichlet–to–Neumann map on the boundary of a bounded open set in Rn\mathbb R^n for the perturbed polyharmonic operator (Δ)m+q(-\Delta)^m +q with qLn2mq\in L^{\frac{n}{2m}}, n>2mn>2m, determines the potential qq 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 L2L^2 and LpL^p spaces. The LpL^p estimates for the special Green function are derived from LpL^p Carleman estimates with linear weights for the polyharmonic operator.

Cite this article

Katsiaryna Krupchyk, Gunther Uhlmann, Inverse boundary problems for polyharmonic operators with unbounded potentials. J. Spectr. Theory 6 (2016), no. 1, pp. 145–183

DOI 10.4171/JST/122