Carleman Estimates and Global Uniqueness for the Inverse Calderón Problem with Partial Data

This monograph addresses the inverse Calderón problem of determining the electrical conductivity of a bounded domain in Rn (n 3) from partial boundary measurements. We establish a global uniqueness theorem for conductivities in C2() given the Dirichlet-to-Neumann map restricted to specific subsets of the boundary. The core of the analysis relies on the construction of Complex Geometric Optics (CGO) solutions to the associated Schrödinger equation via Carleman estimates with limiting Carleman weights. We rigorously derive the Carleman inequality for the semi-classical Laplace operator and utilize it to control the remainder terms in the asymptotic expansion of the CGO solutions. By substituting these solutions into the integral identity derived from the partial data, we prove that the Fourier transform of the difference of the potentials vanishes, thereby implying injectivity of the mapping from conductivity to boundary data. The results generalize the seminal work of Kenig, Sjöstrand, and Uhlmann to a broader class of admissible geometries.