Nonexistence of Courant-type nodal domain bounds for eigenfunctions of the Dirichlet-to-Neumann operator
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| Format: | Preprint |
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2024
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| _version_ | 1866917636539940864 |
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| author | Enciso, Alberto Pistoia, Angela Provenzano, Luigi |
| author_facet | Enciso, Alberto Pistoia, Angela Provenzano, Luigi |
| contents | Given a compact manifold $\mathcal M$ with boundary of dimension $n\geq 3$ and any integers $K$ and $N$, we show that there exists a metric on $\mathcal M$ for which the first $K$ nonconstant eigenfunctions of the Dirichlet-to-Neumann map on $\partial\mathcal M$ have at least $N$ nodal components. This provides a negative answer to the question of whether the number of nodal domains of Dirichlet-to-Neumann eigenfunctions satisfies a Courant-type bound, which has been featured in recent surveys by Girouard and Polterovich [21, Open problem 9] and by Colbois, Girouard, Gordon and Sher [9, Open question 10.14]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_07138 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Nonexistence of Courant-type nodal domain bounds for eigenfunctions of the Dirichlet-to-Neumann operator Enciso, Alberto Pistoia, Angela Provenzano, Luigi Spectral Theory Analysis of PDEs Differential Geometry Given a compact manifold $\mathcal M$ with boundary of dimension $n\geq 3$ and any integers $K$ and $N$, we show that there exists a metric on $\mathcal M$ for which the first $K$ nonconstant eigenfunctions of the Dirichlet-to-Neumann map on $\partial\mathcal M$ have at least $N$ nodal components. This provides a negative answer to the question of whether the number of nodal domains of Dirichlet-to-Neumann eigenfunctions satisfies a Courant-type bound, which has been featured in recent surveys by Girouard and Polterovich [21, Open problem 9] and by Colbois, Girouard, Gordon and Sher [9, Open question 10.14]. |
| title | Nonexistence of Courant-type nodal domain bounds for eigenfunctions of the Dirichlet-to-Neumann operator |
| topic | Spectral Theory Analysis of PDEs Differential Geometry |
| url | https://arxiv.org/abs/2404.07138 |