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New global stability estimates for the Calderón problem in two dimensions

We prove a new global stability estimate for the Gel'fand-Calderón inverse problem on a two-dimensional bounded domain or, more precisely, the inverse boundary value problem for the equation $-Δψ+ v\, ψ= 0$ on $D$, where $v$ is a smooth real-valued potential of conductivity type defined on a bounded planar domain $D$. The principal feature of this estimate is that it shows that the more a potential is smooth, the more its reconstruction is stable, and the stability varies exponentially with respect to the smoothness (in a sense to be made precise). As a corollary we obtain a similar estimate for the Calderón problem for the electrical impedance tomography.