A Transfer Principle for Elliptic Carleman Estimates: Application to Discrete Calderón Stability

Abstract

We develop a transfer principle that lifts continuous elliptic Carleman estimates to mesh-uniform estimates for conforming Galerkin discretizations. The main difficulties are twofold: discrete states generally lack the \(H^2\)-regularity required by the continuous estimate, and the Carleman parameter \(s\) must be controlled relative to the mesh size \(h\). We overcome both by introducing a shifted continuous problem whose Galerkin solution is exactly the conforming reconstruction of a prescribed discrete state. Weighted approximation then transfers the inequality back to the original discrete operator without adding stabilization, and yields the range \(s\le ch^{-4/5}\) for radial weights. We verify the framework for piecewise affine finite elements with consistent or lumped mass on fitted, uniformly shape-regular meshes, and for standard Cartesian finite differences through their finite-element realization. As an application, we study the discrete Calderón problem for bounded complex Schrödinger potentials in dimension \(d\ge3\). Assuming agreement in a fixed connected boundary neighborhood and uniform stability of the discrete Dirichlet problems, we obtain stability from Dirichlet inputs and variational flux observations on independently prescribed, possibly disjoint boundary patches. The resulting estimate in \(H^{-r}\), \(r>d/2\), retains the single logarithmic data term and the \(O(h^{1/2})\) mesh remainder of the full-data estimate.