In this paper we prove that the Cauchy data for the Schrödinger equation in the twodimensional case determines a potential from Lp (for p > 2) uniquely. We also obtain a linear inversion formula for smooth potentials.
In this paper, we discuss an inverse problem of determining a part of a boundary of a bounded domain in the plane. For the determination, we observe both Dirichlet and Neumann data on a subset of a known sub-boundary. We prove various conditional stability estimates according to a priori assumptions on the regularity of unknown sub-boundaries. Our results are: (i) in a general case the distance between two unknown sub-boundaries is conditionally estimated with double logarithmic rate under a priori assumption of C 2 -boundedness. (ii) we can improve stability rates through a single logarithmic rate up to Hölder continuity under the assumption that the sub-boundary is analytic.
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