The inner radius of nodal domains in high dimensions
type: Item
instance of: scholarly article
title: The inner radius of nodal domains in high dimensions
author: Philippe Charron
author: Dan Mangoubi
published in: Advances in Mathematics
publication date: 2024-09-20T00:00:00Z
review text: Let \((M,g)\) be a smooth closed Riemannian manifold of dimension \(d\). Consider on \(M\) an eigenfunction \(u_\lambda\) of the positive Laplace-Beltrami operator \(-\Delta_g\) corresponding to an eigenvalue \(\lambda\). A nodal domain \(\Omega_\lambda\) of \(u_\lambda\) is any connected component of the the set \(\{u_\lambda \neq 0\}\).\N\NIt is well known that there exists a positive constant \(c_{\text{up}} = c_{\text{up}}(M,g)\) independent of \(\lambda\) or \(u_\lambda\) such that every ball of radius bigger than \(c_{\text{up}}(M,g)\lambda^{-1/2}\) contains a zero of \(u_\lambda\) [\textit{J. E. D'Atri} and \textit{W. Ziller}, Naturally reductive metrics and Einstein metrics on compact Lie groups. Providence, RI: American Mathematical Society (AMS) (1979; Zbl 0404.53044); \textit{J. Brüning}, Math. Z. 158, 15--21 (1978; Zbl 0349.58012)]. On the other hand, the Faber-Krahn inequality [\textit{G. Faber}, Münch. Ber. 1923, 169--172 (1923; JFM 49.0342.03); \textit{E. Krahn}, Math. Ann. 94, 97--100 (1925; JFM 51.0356.05)] shows that the volume of every nodal domain \(\Omega_k\) is bounded from below by \(c_{FK}\lambda^{-d/2}\) for some positive constant \(c_{FK} = c_{FK}(M)\).\N\NOne is naturally led to look for the largest positive number \(r\) such that every nodal domain contains a ball of radius \(r\). In two dimensions it is known that once can inscribe a ball of radius \(c\lambda^{-1/2}\) in every nodal domain [\textit{D. Mangoubi}, Can. Math. Bull. 51, No. 2, 249--260 (2008; Zbl 1152.58027)]. This paper is concerned with lower bounds for the inner radius in higher dimensions. In particular, let \((M,g)\) be of dimension \(d\) of at least three. let \(x_{\max}\) be a point where \(|u_\lambda(x_{\max})| = \max_{\Omega_\lambda} |u_\lambda|\). The authors show that the ball of radius \(c_{\text{lo}} \lambda^{-1/2} (\log \lambda)^{-(d-2)/2}\) centered at \(x_{\max}\) is contained in \(\Omega_\lambda\), where \(c_{\text{lo}} = c_{\text{lo}}(M,g)\) is a constant t…
reviewed by: Jacob Shapiro
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 31B15
Mathematics Subject Classification ID: 35K05
Mathematics Subject Classification ID: 35P20
zbMATH DE Number: 7916595
zbMATH Keywords: nodal domains
zbMATH Keywords: inner radius
zbMATH Keywords: Remez inequality
MaRDI profile type: Publication
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DOI: 10.1016/J.AIM.2024.109787
description: scientific article; zbMATH DE number 7916595