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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2510.26331 |
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Sommario:
- 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.