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10 - Neumann Domains on Graphs and Manifolds

Published online by Cambridge University Press:  14 August 2020

Matthias Keller
Affiliation:
Universität Potsdam, Germany
Daniel Lenz
Affiliation:
Universität Potsdam, Germany
Radoslaw K. Wojciechowski
Affiliation:
York College of the City University of New York
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Summary

A Laplacian eigenfunction on a manifold or a metric graph imposes a natural partition of the manifold or the graph. This partition is determined by the gradient vector field of the eigenfunction (on a manifold) or by the extremal points of the eigenfunction (on a graph). The submanifolds (or subgraphs) of this partition are called Neumann domains. Their counterparts are the well-known nodal domains. This paper reviews the subject of Neumann domains, as appears in recent publications and points out some open questions and conjectures. The paper concerns both manifolds and metric graphs and the exposition allows for a comparison between the results obtained for each of them.

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Publisher: Cambridge University Press
Print publication year: 2020

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