The Calderón problem with partial data in two dimensions

Imanuvilov, Oleg¹ ; Uhlmann, Gunther² ; Yamamoto, Masahiro³

¹ Department of Mathematics, Colorado State University, 101 Weber Building, Fort Collins, Colorado 80523
² Department of Mathematics, University of Washington, Seattle, Washington 98195
³ Department of Mathematical Sciences, University of Tokyo, Komaba, Meguro, Tokyo 153, Japan

Journal of the American Mathematical Society, Tome 23 (2010) no. 3, pp. 655-691

Résumé

We prove for a two-dimensional bounded domain that the Cauchy data for the Schrödinger equation measured on an arbitrary open subset of the boundary uniquely determines the potential. This implies, for the conductivity equation, that if we measure the current fluxes at the boundary on an arbitrary open subset of the boundary produced by voltage potentials supported in the same subset, we can uniquely determine the conductivity. We use Carleman estimates with degenerate weight functions to construct appropriate complex geometrical optics solutions to prove the results.

Détail

DOI : 10.1090/S0894-0347-10-00656-9

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