INVERSE BOUNDARY PROBLEMS FOR POLYHARMONIC OPERATORS WITH UNBOUNDED POTENTIALS arXiv:1308.3782v2 [math.AP] 1 Aug 2015 KATSIARYNA KRUPCHYK AND GUNTHER UHLMANN 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 n operator (−∆) m + q with q ∈ L 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. 1. Introduction Let Ω ⊂ R n be a bounded open set with C ∞ boundary, and let (−∆) m , m = n 1, 2, . . . , be a polyharmonic operator. Let q ∈ L 2m (Ω) be a complex valued potential. We shall assume throughout the paper that n > 2m. Let γ be the Dirichlet trace operator, given by m γ : H (Ω) → m−1 Y H m−j−1/2 (∂Ω), γu = (u| ∂Ω , ∂ ν u| ∂Ω , . . . , ∂ ν m−1 u| ∂Ω ), j=0 which is bounded and surjective, see [8, Theorem 9.5, p. 226]. Here and in what follows H s (Ω) and H s (∂Ω), s ∈ R, are the standard L 2 –based Sobolev spaces in Ω and its boundary ∂Ω, respectively, and ν is the exterior unit normal to the boundary. We shall also set H 0 m (Ω) = {u ∈ H m (Ω) : γu = 0}. An application of the Sobolev embedding theorem shows that the operator of multiplication by q is continuous: H 0 m (Ω) → H −m (Ω), and standard arguments, see Appendix A, imply that the operator (−∆) m + q : H 0 m (Ω) → H −m (Ω) = (H 0 m (Ω)) ′ is Fredholm of index zero. Furthermore, the operator in (1.1) has a discrete spectrum. We shall assume throughout the paper that