Multiple tubular excisions and large Steklov eigenvalues
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909131627036672 |
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| author | Brisson, Jade |
| author_facet | Brisson, Jade |
| contents | Given a closed Riemannian manifold $M$ and $b\geq2$ closed connected submanifolds $N_j\subset M$ of codimension at least $2$, we prove that the first non-zero eigenvalue of the domain $Ω_\varepsilon\subset M$ obtained by removing the tubular neighbourhood of size $\varepsilon$ around each $N_j$ tends to infinity as $\varepsilon$ tends to $0$. More precisely, we prove a lower bound in terms of $\varepsilon$, $b$, the geometry of $M$ and the codimensions and the volumes of the submanifolds and an upper bound in terms of $\varepsilon$ and the codimensions of the submanifolds. For eigenvalues of index $k=b\,,b+1\,,\ldots$, we have a stronger result: their order of divergence is $\varepsilon^{-1}$ and their rate of divergence is only depending on $m$ and on the codimensions of the submanifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_00128 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Multiple tubular excisions and large Steklov eigenvalues Brisson, Jade Spectral Theory 58J50, 35P15 Given a closed Riemannian manifold $M$ and $b\geq2$ closed connected submanifolds $N_j\subset M$ of codimension at least $2$, we prove that the first non-zero eigenvalue of the domain $Ω_\varepsilon\subset M$ obtained by removing the tubular neighbourhood of size $\varepsilon$ around each $N_j$ tends to infinity as $\varepsilon$ tends to $0$. More precisely, we prove a lower bound in terms of $\varepsilon$, $b$, the geometry of $M$ and the codimensions and the volumes of the submanifolds and an upper bound in terms of $\varepsilon$ and the codimensions of the submanifolds. For eigenvalues of index $k=b\,,b+1\,,\ldots$, we have a stronger result: their order of divergence is $\varepsilon^{-1}$ and their rate of divergence is only depending on $m$ and on the codimensions of the submanifolds. |
| title | Multiple tubular excisions and large Steklov eigenvalues |
| topic | Spectral Theory 58J50, 35P15 |
| url | https://arxiv.org/abs/2309.00128 |