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The Calderón problem with partial data. (English) Zbl 1127.35079

Summary: We improve an earlier result by A. L. Bukhgeim and G. Uhlmann [Commun. Partial Differ. Equations 27, 653–668 (2002; Zbl 0998.35063)], by showing that in dimension \(n\geq 3\), the knowledge of the Cauchy data for the Schrödinger equation measured on possibly very small subsets of the boundary determines uniquely the potential. We follow the general strategy of Bukhgeim and Uhlmann (loc. cit.) but use a richer set of solutions to the Dirichlet problem. This implies a similar result for the problem of electrical impedance tomography which consists in determining the conductivity of a body by making voltage and current measurements at the boundary.

MSC:

35R30 Inverse problems for PDEs
35J25 Boundary value problems for second-order elliptic equations
35J10 Schrödinger operator, Schrödinger equation
35Q60 PDEs in connection with optics and electromagnetic theory

Citations:

Zbl 0998.35063
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