The second periodic eigenvalue and the Alikakos-Fusco conjecture
type: Item
instance of: scholarly article
title: The second periodic eigenvalue and the Alikakos-Fusco conjecture
zbMATH Open document ID: 0861.34051
author: Vassilis G. Papanicolaou
published in: Journal of Differential Equations
publication date: 1996-11-18T00:00:00Z
review text: The author considers the periodic eigenvalue problem \(-u''+qu=\lambda u\), \(u(0)= u(2\pi)\), \(u'(0)= u'(2\pi)\), where \(q\in L^1(0,2\pi)\) and \(\int_0^{2\pi} q(x)dx=0\). It is proved that the second eigenvalue \(\lambda_1\) satisfies \(\lambda_1\leq 1\) and \(\lambda_1=1\) if and only if \(q=0\), for the following potentials: (i) \(q(x)= a_1\cos x+\sum_{n\geq 2}(a_n\cos nx+b_n\sin nx)\) and \(a_2\geq 0\) or \(a_1=0\); (ii) \(q(x)= \varepsilon Q(x)\), where \(Q\) is fixed and \(|\varepsilon|\) is small enough or sufficiently large. In particular, the one-dimensional Alikakos-Fusco conjecture is verified for these potentials.
Mathematics Subject Classification ID: 34L15
Mathematics Subject Classification ID: 37A30
zbMATH DE Number: 946931
zbMATH Keywords: periodic eigenvalue problem
zbMATH Keywords: second eigenvalue
zbMATH Keywords: one-dimensional Alikakos-Fusco conjecture
reviewed by: Georgi E. Karadzhov
MaRDI profile type: Publication
full work available at URL: jdeq.1996.0146
OpenAlex ID: W2036826856
Wikidata QID: Q123361718
DOI: 10.1006/JDEQ.1996.0146
description: scientific article; zbMATH DE number 946931