Complete spectrum of the Robin eigenvalue problem on the ball
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911291461861376 |
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| author | Dai, Guowei Sun, Yingxin |
| author_facet | Dai, Guowei Sun, Yingxin |
| contents | We investigate the following Robin eigenvalue problem \begin{equation*} \left\{ \begin{array}{ll} -Δu=μu\,\, &\text{in}\,\, B,\\ \partial_\texttt{n} u+αu=0 &\text{on}\,\, \partial B \end{array} \right. \end{equation*} on the unit ball of $\mathbb{R}^N$. We obtain the complete spectral structure of this problem. In particular, for $α>0$, the first eigenvalue is $k_{ν,1}^2$ and the second eigenvalue is $k_{ν+1,1}^2$, where $k_{ν+l,m}$ is the $m$th positive zero of $kJ_{ν+l+1}(k)-(α+l) J_{ν+l}(k)$. Moreover, when $α\in(-l,1-l)$ with any $l\in \mathbb{N}$, one has $l$ negative (strictly increasing) eigenvalues $-\widehat{k}_{ν+i,1}^2$ with $i\in\{0,\ldots,l-1\}$ where $\widehat{k}_{ν+l,1}$ denotes the unique zero of $αI_{ν+l}(k)+lI_{ν+l}(k)+kI_{ν+l+1}(k)$; while, for $α=-l$, besides $l$ negative (increasing) eigenvalues, $0$ is also an eigenvalue. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_26331 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Complete spectrum of the Robin eigenvalue problem on the ball Dai, Guowei Sun, Yingxin Analysis of PDEs Spectral Theory 34L20, 34L10, 35J05, 35P10, 35P20 We investigate the following Robin eigenvalue problem \begin{equation*} \left\{ \begin{array}{ll} -Δu=μu\,\, &\text{in}\,\, B,\\ \partial_\texttt{n} u+αu=0 &\text{on}\,\, \partial B \end{array} \right. \end{equation*} on the unit ball of $\mathbb{R}^N$. We obtain the complete spectral structure of this problem. In particular, for $α>0$, the first eigenvalue is $k_{ν,1}^2$ and the second eigenvalue is $k_{ν+1,1}^2$, where $k_{ν+l,m}$ is the $m$th positive zero of $kJ_{ν+l+1}(k)-(α+l) J_{ν+l}(k)$. Moreover, when $α\in(-l,1-l)$ with any $l\in \mathbb{N}$, one has $l$ negative (strictly increasing) eigenvalues $-\widehat{k}_{ν+i,1}^2$ with $i\in\{0,\ldots,l-1\}$ where $\widehat{k}_{ν+l,1}$ denotes the unique zero of $αI_{ν+l}(k)+lI_{ν+l}(k)+kI_{ν+l+1}(k)$; while, for $α=-l$, besides $l$ negative (increasing) eigenvalues, $0$ is also an eigenvalue. |
| title | Complete spectrum of the Robin eigenvalue problem on the ball |
| topic | Analysis of PDEs Spectral Theory 34L20, 34L10, 35J05, 35P10, 35P20 |
| url | https://arxiv.org/abs/2510.26331 |