On uniqueness in the inverse conductivity problem with local data

The inverse condictivity problem with many boundary measurements consists of recovery of conductivity coefficient a (principal part) of an elliptic equation in a domain Ω ⊂ R, n = 2, 3 from the Neumann data given for all Dirichlet data (Dirichlet-to-Neumann map). Calderon [5] proposed the idea of using complex exponential solutions to demonstrate uniqueness in the linearized inverse condictivity problem. Complex exponential solutions of elliptic equations have been introduced by Faddeev [7] for needs of inverse scattering theory. Sylvester and Uhlmann in their fundamental paper [19] attracted ideas from geometrical optics, constructed almost complex exponential solutions for the Schrödinger operator, and proved global uniqueness of a ( and of potential c in the Schrödinger equation) in the three-dimensional case. In the two-dimensional case the inverse conductivity problem is less overdetermined, and the Sylvester and Uhlmann method is not applicable, but one enjoys advantages of the methods of inverse scattering and of theory of complex variables. Using these methods Nachman [17] demonstrated uniqueness of a ∈ C(Ω̄) and Astala and Päivärinta [1] showed uniqueness of a ∈ L∞(Ω) which is a final result in the inverse conductivity problem in R with many measurements from the whole boundary. There is a known hypothesis (see for example, [11], Problem 5.3, [14], [20]) that the Dirichlet-to-Neumann map given at any (nonvoid open) part Γ of the boundary also uniquely determines conductivity coefficient or potential in the Schrödinger equation. This local boundary measurements model important