Partial Data for the Calderon Problem in Two Dimensions

TL;DR

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

Contribution

It establishes a new uniqueness result for the Calderon problem with partial data in two dimensions, extending previous knowledge.

Findings

Unique determination of conductivity from partial boundary data.

Application of Carleman estimates with degenerate weights.

Construction of complex geometrical optics solutions for the problem.

Abstract

We show in two dimensions that measuring Dirichlet data for the conductivity equation on an open subset of the boundary and, roughly speaking, Neumann data in slightly larger set than the complement uniquely determines the conductivity on a simply connected domain. The proof is reduced to show a similar result for the Schr\"odinger equation. Using Carleman estimates with degenerate weights we construct appropriate complex geometrical optics solutions to prove the results.