Homology of spectral minimal partitions
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866914827704729600 |
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| author | Berkolaiko, Gregory Canzani, Yaiza Cox, Graham Marzuola, Jeremy L. |
| author_facet | Berkolaiko, Gregory Canzani, Yaiza Cox, Graham Marzuola, Jeremy L. |
| contents | A spectral minimal partition of a manifold is its decomposition into disjoint open sets that minimizes a spectral energy functional. It is known that bipartite spectral minimal partitions coincide with nodal partitions of Courant-sharp Laplacian eigenfunctions. However, almost all minimal partitions are non-bipartite. To study those, we define a modified Laplacian operator and prove that the nodal partitions of its Courant-sharp eigenfunctions are minimal within a certain topological class of partitions. This yields new results in the non-bipartite case and recovers the above known result in the bipartite case. Our approach is based on tools from algebraic topology, which we illustrate by a number of examples where the topological types of partitions are characterized by relative homology. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_04225 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Homology of spectral minimal partitions Berkolaiko, Gregory Canzani, Yaiza Cox, Graham Marzuola, Jeremy L. Analysis of PDEs Algebraic Topology Spectral Theory A spectral minimal partition of a manifold is its decomposition into disjoint open sets that minimizes a spectral energy functional. It is known that bipartite spectral minimal partitions coincide with nodal partitions of Courant-sharp Laplacian eigenfunctions. However, almost all minimal partitions are non-bipartite. To study those, we define a modified Laplacian operator and prove that the nodal partitions of its Courant-sharp eigenfunctions are minimal within a certain topological class of partitions. This yields new results in the non-bipartite case and recovers the above known result in the bipartite case. Our approach is based on tools from algebraic topology, which we illustrate by a number of examples where the topological types of partitions are characterized by relative homology. |
| title | Homology of spectral minimal partitions |
| topic | Analysis of PDEs Algebraic Topology Spectral Theory |
| url | https://arxiv.org/abs/2406.04225 |