The complete Dirichlet-to-Neumann map for differential forms

TL;DR

This paper develops an invariant, comprehensive framework for the Dirichlet-to-Neumann map for differential forms on manifolds, linking it to topological invariants and unifying different existing approaches.

Contribution

It introduces an invariant definition of the complete Dirichlet-to-Neumann map for differential forms using operators {\Phi} and {\Psi}, connecting various prior formulations.

Findings

Betti numbers are determined by {\Phi}.

{\Psi} defines a chain complex related to cohomology.

Unified approach to Dirichlet-to-Neumann maps for differential forms.

Abstract

The Dirichlet-to-Neumann map for differential forms on a Riemannian manifold with boundary is a generalization of the classical Dirichlet-to-Neumann map which arises in the problem of Electrical Impedance Tomography. We synthesize the two different approaches to defining this operator by giving an invariant definition of the complete Dirichlet-to-Neumann map for differential forms in terms of two linear operators {\Phi} and {\Psi}. The pair ({\Phi}, {\Psi}) is equivalent to Joshi and Lionheart's operator {\Pi} and determines Belishev and Sharafutdinov's operator {\Lambda}. We show that the Betti numbers of the manifold are determined by {\Phi} and that {\Psi} determines a chain complex whose homologies are explicitly related to the cohomology groups of the manifold.