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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.

We recall here some theoretical results of Helffer et al. [Ann. Inst. H. Poincaré Anal. Non Linéaire (2007) doi:10.1016/j.anihpc.2007.07.004] about minimal partitions and propose numerical computations to check some of their published or unpublished conjectures and exhibit new ones.

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10 - Neumann Domains on Graphs and Manifolds

Summary

Numerical simulations for nodal domainsand spectral minimal partitions

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2 results

10 - Neumann Domains on Graphs and Manifolds

Summary

Numerical simulations for nodal domainsand spectral minimal partitions