Inverse boundary problems for polyharmonic operators with unbounded potentials

2013/08/17 by Krupchyk, Katsiaryna, Uhlmann, Gunther
#31B30 #35J40 #35R30 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.1308.3782

Abstract

We show that the knowledge of the Dirichlet-to-Neumann map on the boundary of a bounded open set in Rⁿ for the perturbed polyharmonic operator (-Δ)^m +q with q∈ L^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 L² 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.

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