Stability Estimates for the Inverse Conductivity Problem with Partial Boundary Data using Carleman Inequalities

We investigate the stability of the inverse conductivity problem, specifically the determination of an isotropic electrical conductivity distribution within a bounded domain in Rn (n 3) from the Dirichlet-to-Neumann map measured on a subset of the boundary. This problem, known as the Calderón problem with partial data, is ill-posed, necessitating rigorous stability estimates to quantify the continuous dependence of the conductivity on the boundary measurements. By employing Carleman inequalities with limiting Carleman weights, we derive a logarithmic stability estimate for conductivities in Hs() for sufficiently large s. The analysis relies on the reduction of the conductivity equation to a Schrödinger-type equation via the Liouville transformation and the construction of Complex Geometric Optics (CGO) solutions that vanish on the inaccessible part of the boundary. We establish that if the difference between the Dirichlet-to-Neumann maps of two conductivities is small in the operator norm, the L-norm of the difference between the potentials (and hence the conductivities) is controlled by a logarithmic function of the data error, provided the conductivities satisfy a priori regularity bounds.