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

New global stability estimates for the Calder\'on problem in two dimensions