Two disks maximize the third Robin eigenvalue: positive parameters

doi:10.1007/s40316-025-00254-x

Two disks maximize the third Robin eigenvalue: positive parameters

2025-09-01

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DOI：10.1007/s40316-025-00254-x

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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 [-4 pi,4 pi].\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[-4\pi ,4\pi ].$$\end{document} This sharp inequality was known previously only for negative parameters in [-4 pi,0],\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[-4\pi ,0],$$\end{document} 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 (0,4 pi]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(0,4\pi ]$$\end{document} by means of a degree-theoretic approach suggested by Karpukhin and Stern that yields suitably orthogonal trial functions. La troisi & egrave;me valeur propre du Laplacien de Robin sur un domaine simplement connexe du plan, dont l'aire est donn & eacute;e, est born & eacute;e sup & eacute;rieurement par la valeur propre correspondante d'une union disjointe de deux disques & eacute;gaux, pour des param & egrave;tres de Robin dans [-4 pi,4 pi].\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[-4\pi ,4\pi ].$$\end{document} Cette in & eacute;galit & eacute; optimale n'& eacute;tait auparavant connue que pour des param & egrave;tres n & eacute;gatifs dans [-4 pi,0],\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[-4\pi ,0],$$\end{document} par Girouard et Laugesen. Leur d & eacute;monstration & eacute;choue pour des param & egrave;tres de Robin positifs parce que la partie radiale de la deuxi & egrave;me fonction propre sur le disque n'est pas monotone. Cette difficult & eacute; est surmont & eacute;e pour des param & egrave;tres dans (0,4 pi]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(0,4\pi ]$$\end{document} en utilisant une approche via la th & eacute;orie du degr & eacute;, sugg & eacute;r & eacute;e par Karpukhin et Stern, qui produit des fonctions tests convenablement orthogonales.

Keywords:

Robin

Neumann

Vibrating membrane

Conformal mapping

Two disks maximize the third Robin eigenvalue: positive parameters

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