On the Calderón problem with piecewise polynomial anisotropic conductivities and many-flat-face interfaces

Abstract

We prove uniqueness for a finite-dimensional anisotropic Calderón problem in dimension \(n\ge 3\). The conductivity is a symmetric uniformly elliptic matrix field which is cellwise polynomial with respect to a known finite partition, and it may be discontinuous across interfaces. Under a geometric assumption allowing the cells to be reached successively through sufficiently many flat interface patches, the local Dirichlet-to-Neumann map on an initial boundary patch determines all polynomial pieces. The proof combines a one-cell recovery result from flat-face boundary data with a layer-stripping argument, using Runge approximation and unique continuation to propagate the data across already recovered cells. As a consequence of a finite-dimensional analytic stability theorem, the recovery is Hölder stable on compact admissible parameter sets.