The Calderón problem for piecewise polynomial anisotropic conductivities on partitions with many flat faces

Abstract

We prove uniqueness for a finite-dimensional anisotropic Calderón problem in dimensions $n\ge 3$. The unknown conductivity is a symmetric uniformly elliptic matrix field whose restriction to each cell of a known finite partition is polynomial; jumps across cell interfaces are allowed. We assume that the cells can be ordered so that each successive cell is accessible through sufficiently many flat faces. Under this assumption, the local Dirichlet-to-Neumann map on an exterior boundary patch determines every polynomial piece. The proof first recovers one polynomial matrix from its boundary invariants on several flat faces and then propagates the required local data through the recovered cells by unique continuation and Runge approximation. A general finite-dimensional analytic stability theorem also gives Hölder stability on compact admissible parameter sets.