Complete spectrum of the Robin eigenvalue problem on the ball

Abstract

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 $\hat{I}\pm >0$, the first eigenvalue is $k_{\hat{I}½,1}^2$ and the second eigenvalue is $k_{\hat{I}½+1,1}^2$, where $k_{\hat{I}½+l,m}$ is the $m$th positive zero of $k{J}_{\hat{I}½+l+1}(k)-(\hat{I}\pm +l)J_{\hat{I}½+l}(k)$. Moreover, when $\hat{I}\pm \in (-l,1-l)$ with any $l\in \mathbb{N}$, one has $l$ negative (strictly increasing) eigenvalues $-{\hat{k}}_{\hat{I}½+i,1}^2$ with $i\in \{0,\dots ,l-1\}$ where ${\hat{k}}_{\hat{I}½+l,1}$ denotes the unique zero of $\hat{I}\pm I_{\hat{I}½+l}(k)+l{I}_{\hat{I}½+l}(k)+k{I}_{\hat{I}½+l+1}(k)$; while, for $\hat{I}\pm =-l$, besides $l$ negative (increasing) eigenvalues, $0$ is also an eigenvalue.