The local Calderon problem and the determination at the boundary of the conductivity

We discuss the inverse problem of determining the, possibly anisotropic, conductivity of a body $\mathrm{\Omega }\subset {\mathbb{R}}^n$ when the so–called Dirichlet-to-Neumann map is locally given on a non empty portion $\mathrm{\Gamma }$ of the boundary $\mathrm{\partial }\mathrm{\Omega }$. We extend results of uniqueness and stability at the boundary, obtained by the same authors in SIAM J. Math. Anal. 33 (2001), no. 1, 153–171, where the Dirichlet-to-Neumann map was given on all of $\mathrm{\partial }\mathrm{\Omega }$ instead. We also obtain a pointwise stability result at the boundary among the class of conductivities which are continuous at some point $y\in \mathrm{\Gamma }$. Our arguments also apply when the local Neumann-to-Dirichlet map is available.