Complete spectrum of the Robin eigenvalue problem on the ball

TL;DR

This work analyzes the Robin eigenvalue problem for the Laplacian on the unit ball in $\mathbb{R}^N$, aiming to determine its complete spectrum and the influence of the boundary parameter $\alpha$. It employs separation of variables and Bessel-function theory to derive explicit formulas for eigenvalues and eigenfunctions, revealing a full classification into positive, zero, and negative spectra depending on $\alpha$. The key contributions include explicit expressions for the first two eigenvalues when $\alpha>0$, a complete description of the spectrum $\mu_{l,m}$ indexed by angular momentum and radial index, and precise conditions determining the number of negative eigenvalues, with detailed asymptotics as $\alpha$ varies. These results extend known 1D insights to higher dimensions and provide a concrete basis for addressing related spectral conjectures and optimization problems on balls.

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 $α>0$, 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.

Figures (2)

• Figure 1: The schematic diagram of $\alpha_+$ (solid lines) and $\alpha_-$ (dotted lines).
• Figure 2: The schematic diagram of $h_{\nu+l}(k)$.