Global uniqueness from partial Cauchy data in two dimensions

TL;DR

This paper proves that in two dimensions, boundary measurements of the Schrödinger and conductivity equations uniquely determine the potential and conductivity, respectively, using Carleman estimates and complex geometrical optics solutions.

Contribution

It establishes the uniqueness of potential and conductivity determination from partial boundary data in two dimensions, extending previous results to arbitrary open boundary subsets.

Findings

Unique determination of potential from partial boundary data for Schrödinger equation

Unique determination of conductivity from boundary flux measurements

Application of Carleman estimates with degenerate weights

Abstract

We prove for a two dimensional bounded domain that the Cauchy data for the Schroedinger equation measured on an arbitrary open subset of the boundary determines uniquely the potential. This implies, for the conductivity equation, that if we measure the current fluxes at the boundary on an arbitrary open subset of the boundary produced by voltage potentials supported in the same subset, we can determine uniquely the conductivity. We use Carleman estimates with degenerate weight functions to construct appropriate complex geometrical optics solutions to prove the results.