Complete spectrum of the Robin eigenvalue problem on the ball

TL;DR

This paper provides a complete spectral analysis of the Robin eigenvalue problem on a unit ball, detailing the eigenvalues' structure and zeros of related Bessel functions for various boundary parameter values.

Contribution

It derives the full spectrum of the Robin problem on the ball, explicitly characterizing eigenvalues in terms of zeros of Bessel functions and their modifications for different boundary conditions.

Findings

First eigenvalue is k_{ν,1}^2 for α>0

Second eigenvalue is k_{ν+1,1}^2 for α>0

Multiple negative eigenvalues for certain α ranges

Abstract

We investigate the following Robin eigenvalue problem \begin{equation*} \left\{ \begin{array}{ll} -\Delta u=\mu u\,\, &\text{in}\,\, B,\\ \partial_\texttt{n} u+\alpha u=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 $\alpha>0$, the first eigenvalue is $k_{\nu,1}^2$ and the second eigenvalue is $k_{\nu+1,1}^2$, where $k_{\nu+l,m}$ is the $m$th positive zero of $kJ_{\nu+l+1}(k)-(\alpha+l) J_{\nu+l}(k)$. Moreover, when $\alpha\in(-l,1-l)$ with any $l\in \mathbb{N}$, one has $l$ negative (strictly increasing) eigenvalues $-\widehat{k}_{\nu+i,1}^2$ with $i\in\{0,\ldots,l-1\}$ where $\widehat{k}_{\nu+l,1}$ denotes the unique zero of $\alpha I_{\nu+l}(k)+lI_{\nu+l}(k)+kI_{\nu+l+1}(k)$; while, for $\alpha=-l$, besides $l$ negative (increasing) eigenvalues, $0$ is also an eigenvalue.