Hausdorff measure bounds for nodal sets of Steklov eigenfunctions
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866909207330029568 |
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| author | Decio, Stefano |
| author_facet | Decio, Stefano |
| contents | We study nodal sets of Steklov eigenfunctions in a bounded domain with $\mathcal{C}^2$ boundary. Our first result is a lower bound for the Hausdorff measure of the nodal set: we show that for $u_λ$ a Steklov eigenfunction, with eigenvalue $λ\neq 0$, $\mathcal{H}^{d-1}(\{u_λ=0\})\geq c_Ω$, where $c_Ω$ is independent of $λ$. We also prove an almost sharp upper bound, namely $\mathcal{H}^{d-1}(\{u_λ=0\})\leq C_Ωλ\log(λ+e)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2104_10275 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Hausdorff measure bounds for nodal sets of Steklov eigenfunctions Decio, Stefano Analysis of PDEs Spectral Theory We study nodal sets of Steklov eigenfunctions in a bounded domain with $\mathcal{C}^2$ boundary. Our first result is a lower bound for the Hausdorff measure of the nodal set: we show that for $u_λ$ a Steklov eigenfunction, with eigenvalue $λ\neq 0$, $\mathcal{H}^{d-1}(\{u_λ=0\})\geq c_Ω$, where $c_Ω$ is independent of $λ$. We also prove an almost sharp upper bound, namely $\mathcal{H}^{d-1}(\{u_λ=0\})\leq C_Ωλ\log(λ+e)$. |
| title | Hausdorff measure bounds for nodal sets of Steklov eigenfunctions |
| topic | Analysis of PDEs Spectral Theory |
| url | https://arxiv.org/abs/2104.10275 |