Cheeger cuts and Robin spectral minimal partitions of metric graphs
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909230827569152 |
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| author | Kennedy, James B. Ribeiro, João P. |
| author_facet | Kennedy, James B. Ribeiro, João P. |
| contents | We study partition problems based on two ostensibly different kinds of energy functionals defined on $k$-partitions of metric graphs: Cheeger-type functionals whose minimisers are the $k$-Cheeger cuts of the graph, and the corresponding values are the $k$-Cheeger constants of the graph; and functionals built using the first eigenvalue of the Laplacian with positive, i.e. absorbing, Robin (delta) vertex conditions at the boundary of the partition elements. We prove existence of minimising $k$-partitions, $k \geq 2$, for both these functionals. We also show that, for each $k \geq 2$, as the Robin parameter $α\to 0$, up to a renormalisation the spectral minimal Robin energy converges to the $k$-Cheeger constant. Moreover, up to a subsequence, the Robin spectral minimal $k$-partitions converge in a natural sense to a $k$-Cheeger cut of the graph. Finally, we show that as $α\to \infty$ there is convergence in a similar sense to the corresponding Dirichlet minimal energy and partitions.
It is strongly expected that similar results hold on general (smooth, bounded) Euclidean domains and manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_02701 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Cheeger cuts and Robin spectral minimal partitions of metric graphs Kennedy, James B. Ribeiro, João P. Spectral Theory We study partition problems based on two ostensibly different kinds of energy functionals defined on $k$-partitions of metric graphs: Cheeger-type functionals whose minimisers are the $k$-Cheeger cuts of the graph, and the corresponding values are the $k$-Cheeger constants of the graph; and functionals built using the first eigenvalue of the Laplacian with positive, i.e. absorbing, Robin (delta) vertex conditions at the boundary of the partition elements. We prove existence of minimising $k$-partitions, $k \geq 2$, for both these functionals. We also show that, for each $k \geq 2$, as the Robin parameter $α\to 0$, up to a renormalisation the spectral minimal Robin energy converges to the $k$-Cheeger constant. Moreover, up to a subsequence, the Robin spectral minimal $k$-partitions converge in a natural sense to a $k$-Cheeger cut of the graph. Finally, we show that as $α\to \infty$ there is convergence in a similar sense to the corresponding Dirichlet minimal energy and partitions. It is strongly expected that similar results hold on general (smooth, bounded) Euclidean domains and manifolds. |
| title | Cheeger cuts and Robin spectral minimal partitions of metric graphs |
| topic | Spectral Theory |
| url | https://arxiv.org/abs/2310.02701 |