Two disks maximize the third Robin eigenvalue: positive parameters

TL;DR

This work proves a sharp upper bound for the third Robin eigenvalue $\\lambda_3$ on simply-connected planar domains of fixed area, showing it is maximized (in the degenerating limit) by two equal disks for all Robin parameters in $[-4\\pi,4\\pi]$ and extending the positive-range result $(0,4\\pi]$ via a degree-theoretic method. The authors build a four-parameter family of trial functions $u_{w,C}$ from disk eigenfunctions through conformal maps, hyperbolic caps, and fold/cap transformations, and then prove the existence of a trial function orthogonal to the first two eigenfunctions using a vanishing vector-field argument grounded in degree theory for reflection-symmetric maps on $\mathbb{S}^3$. A key technical contribution is a streamlined degree-theory lemma (special case of Karpukhin-Stern) to certify the required orthogonality, enabling extension of Girouard-Laugesen’s approach to positive Robin parameters. The result resolves an open problem and provides a robust framework for sharp spectral bounds via conformal and topological methods with potential applications to related eigenvalue optimization problems. The interplay of Möbius transformations, hyperbolic caps, and degree theory highlights a powerful blend of complex analysis, geometry, and topology in spectral optimization.

Abstract

The third eigenvalue of the Robin Laplacian on a simply-connected planar domain of given area is bounded above by the corresponding eigenvalue of a disjoint union of two equal disks, for Robin parameters in $[-4π,4π]$. This sharp inequality was known previously only for negative parameters in $[-4π,0]$, by Girouard and Laugesen. Their proof fails for positive Robin parameters because the second eigenfunction on a disk has non-monotonic radial part. This difficulty is overcome for parameters in $(0,4π]$ by means of a degree-theoretic approach suggested by Karpukhin and Stern that yields suitably orthogonal trial functions.

Figures (5)

• Figure 1: The hyperbolic cap $C=C_{p,t}$ is the image of the half-disk $C_{p,0}$ under the Möbius transform $M_{-pt}$. As shown in the diagram, positive $t$ values correspond to caps larger than a half-disk.
• Figure 2: The trial function $u_{w,C}$ on $\Omega$ is constructed by precomposing the (complex-valued) second Robin eigenfunction $v$ on the disk with four transformations: map conformally from $\Omega$ to the disk, fold onto the cap $C$, expand the cap to the whole disk, apply a Möbius map of the disk, and finally evaluate $v$. The cap $C$ and Möbius parameter $w$ will be chosen to ensure orthogonality of the trial function to the first and second Robin eigenfunctions on $\Omega$.
• Figure 3: Steps 1 and 2 --- extension of $\phi$ to $\phi_2$. We consider the mapping $\phi$ on the sphere ${\mathbb S}^2$ (shown here as a circle) and extend to the ball ${\mathbb B}^3$, mapping it into ${\mathbb R}^4$ while maintaining reflection symmetry. The extended map $\phi_2$ equals the identity on the sets shown in blue, namely the $3$-ball of radius $1/2$ and the lower dimensional ball ${\mathbb B}^2$ represented along the horizontal axis. The image of ${\mathbb B}^3$ generally lies outside ${\mathbb R}^3$, as indicated by the deformed circle on the right.
• Figure 4: Step 3 --- extension of $\phi_2$ to $\phi_3$. We consider the mapping $\phi_2$ on the sphere ${\mathbb S}^3$ and extend to the ball ${\mathbb B}^4$ while maintaining reflection symmetry. The extended map $\phi_3$ is the identity on the $4$-ball of radius $1/2$, shown in blue, and on the lower dimensional ball ${\mathbb B}^2$ represented along the horizontal axis.
• Figure 5: Plot of the radial part $g(r)$ of the second Robin eigenfunction $g(r) e^{i \theta}$ on the unit disk, for a range of $\alpha$-values. In terms of the $J_1$ Bessel function, $g(r)=(\text{const.})J_1(r\sqrt{\lambda_2(\mathbb{D};\alpha)}\,)$ when $\alpha>-1$. Credit:FreiLauS2020.

Theorems & Definitions (10)

• Theorem 1.1: Third Robin eigenvalue is maximal for two disks
• Lemma 1: Continuous dependence of trial function; GirLau2021
• Lemma 2: Extension of trial function to large caps; GirLau2021
• Proposition 1: Vanishing of the vector field
• proof
• Theorem 5.1: Degree of reflection-symmetric map between $3$-spheres
• proof
• Lemma 3: Reflection symmetry and degrees on the half-annuli
• proof
• Proposition 2: Second Robin eigenfunctions of the disk