Neumann Domains on Graphs and Manifolds

2018/05/19 by Lior Alon, Alon, Lior, Ram Band +5 · 1 citation
Mathematics · #34B45 #35Pxx #57M20 #81Q35 #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Mathematical Physics (math-ph) #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.1805.07612

openalex publication_date 2018/05/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The nodal set of a Laplacian eigenfunction forms a partition of the underlying manifold or graph. Another natural partition is based on the gradient vector field of the eigenfunction (on a manifold) or on the extremal points of the eigenfunction (on a graph). The submanifolds (or subgraphs) of this partition are called Neumann domains. This paper reviews the subject, as appears in a few recent works 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.

Cited by

• Spectral flow and Robin domains on metric graphs

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