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The Neumann-to-Dirichlet map in two dimensions. (English) Zbl 1321.35024

Authors’ abstract: For the two-dimensional Schrödinger equation in a bounded domain, we prove uniqueness of the determination of potentials in \(W^1_p(\Omega)\), \(p>2\) in the case where we apply all possible Neumann data supported on an arbitrarily non-empty open set \(\tilde{\Gamma}\) of the boundary and observe the corresponding Dirichlet data on \(\tilde{\Gamma}\). An immediate consequence is that one can uniquely determine a conductivity in \(W^3_p(\Omega)\) with \(p>2\) by measuring the voltage on an open subset of the boundary corresponding to a current supported in the same set.

MSC:

35J25 Boundary value problems for second-order elliptic equations
35J10 Schrödinger operator, Schrödinger equation
35Q60 PDEs in connection with optics and electromagnetic theory
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