Reconstruction in the Calderon Problem with Partial Data

TL;DR

This paper proves a uniqueness result for recovering a coefficient in an elliptic PDE from partial boundary measurements, introducing new Green's functions and integral equation methods that could enhance inverse problem solutions.

Contribution

It provides a constructive proof of uniqueness in the Calderon problem with partial data, using novel Green's functions and boundary integral equations.

Findings

Constructed a family of solutions with boundary traces computed from partial data.

Developed new Green's functions for the Laplacian relevant to inverse problems.

Established a constructive approach to recover ta(x) from limited boundary measurements.

Abstract

We consider the problem of recovering the coefficient \sigma(x) of the elliptic equation \grad \cdot(\sigma \grad u)=0 in a body from measurements of the Cauchy data on possibly very small subsets of its surface. We give a constructive proof of a uniqueness result by Kenig, Sj\"ostrand, and Uhlmann. We construct a uniquely specified family of solutions such that their traces on the boundary can be calculated by solving an integral equation which involves only the given partial Cauchy data. The construction entails a new family of Green's functions for the Laplacian, and corresponding single layer potentials, which may be of independent interest.