Stability of spectral partitions with corners
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910683867643904 |
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| author | Berkolaiko, Gregory Canzani, Yaiza Cox, Graham Kuchment, Peter Marzuola, Jeremy L. |
| author_facet | Berkolaiko, Gregory Canzani, Yaiza Cox, Graham Kuchment, Peter Marzuola, Jeremy L. |
| contents | A spectral minimal partition of a manifold is a decomposition into disjoint open sets that minimizes a spectral energy functional. While it is known that bipartite minimal partitions correspond to nodal partitions of Courant-sharp Laplacian eigenfunctions, the non-bipartite case is much more challenging. In this paper, we unify the bipartite and non-bipartite settings by defining a modified Laplacian operator and proving that the nodal partitions of its eigenfunctions are exactly the critical points of the spectral energy functional. Moreover, we prove that the Morse index of a critical point equals the nodal deficiency of the corresponding eigenfunction. Some striking consequences of our main result are: 1) in the bipartite case, every local minimum of the energy functional is in fact a global minimum; 2) in the non-bipartite case, every local minimum of the energy functional minimizes within a certain topological class of partitions. Our results are valid for partitions with non-smooth boundaries; this introduces considerable technical challenges, which are overcome using delicate approximation arguments in the Sobolev space $H^{1/2}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_02378 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Stability of spectral partitions with corners Berkolaiko, Gregory Canzani, Yaiza Cox, Graham Kuchment, Peter Marzuola, Jeremy L. Analysis of PDEs Spectral Theory A spectral minimal partition of a manifold is a decomposition into disjoint open sets that minimizes a spectral energy functional. While it is known that bipartite minimal partitions correspond to nodal partitions of Courant-sharp Laplacian eigenfunctions, the non-bipartite case is much more challenging. In this paper, we unify the bipartite and non-bipartite settings by defining a modified Laplacian operator and proving that the nodal partitions of its eigenfunctions are exactly the critical points of the spectral energy functional. Moreover, we prove that the Morse index of a critical point equals the nodal deficiency of the corresponding eigenfunction. Some striking consequences of our main result are: 1) in the bipartite case, every local minimum of the energy functional is in fact a global minimum; 2) in the non-bipartite case, every local minimum of the energy functional minimizes within a certain topological class of partitions. Our results are valid for partitions with non-smooth boundaries; this introduces considerable technical challenges, which are overcome using delicate approximation arguments in the Sobolev space $H^{1/2}$. |
| title | Stability of spectral partitions with corners |
| topic | Analysis of PDEs Spectral Theory |
| url | https://arxiv.org/abs/2411.02378 |