An inverse problem of Calderon type with partial data

TL;DR

This paper addresses a generalized Calderón inverse problem with partial boundary data, demonstrating unique determination and reconstruction of the Dirichlet operator for anisotropic conductivities in arbitrary dimensions.

Contribution

It extends Calderón problem results to anisotropic Lipschitz conductivities with partial data, showing unique determination and reconstruction of the Dirichlet operator.

Findings

Unique determination of the Dirichlet operator up to unitary equivalence.

Reconstruction of the Dirichlet operator from residuals of the Dirichlet-to-Neumann map.

Applicability to arbitrary space dimensions $n \\geq 2$.

Abstract

A generalized variant of the Calder\'on problem from electrical impedance tomography with partial data for anisotropic Lipschitz conductivities is considered in an arbitrary space dimension $n \geq 2$. The following two results are shown: (i) The selfadjoint Dirichlet operator associated with an elliptic differential expression on a bounded Lipschitz domain is determined uniquely up to unitary equivalence by the knowledge of the Dirichlet-to-Neumann map on an open subset of the boundary, and (ii) the Dirichlet operator can be reconstructed from the residuals of the Dirichlet-to-Neumann map on this subset.