  We prove an optimal stability estimate for Electrical Impedance Tomography
with local data, in the case when the conductivity is precisely known on a
neighborhood of the boundary. The main novelty here is that we provide a rather
general method which enables to obtain the H\"older dependence of a global
Dirichlet to Neumann map from a local one on a larger domain when, in the layer
between the two boundaries, the coefficient is known.

Analysis of PDEs