All periodic minimizers are unstable
type: Item
instance of: scholarly article
title: All periodic minimizers are unstable
zbMATH Open document ID: 1155.34022
author: Antonio J. Ureña
published in: Archiv der Mathematik
publication date: 2008-09-09T00:00:00Z
review text: The paper deals with the action functional associated to a Lagrangian acting on scalar-valued periodic functions. The problem of the instability of (local) minimizers goes back to Poincaré. In 1935, {C. Carathéodory} showed [cf. Calculus of variations and partial differential equations of the first order, Chelsea, New York (1982; Zbl 0505.49001)] that the nondegenerate periodic minimizers of this functional are always unstable. \textit{E. N. Dancer} and \textit{R. Ortega} [J. Dyn. Differ. Equ. 6, 631--637 (1994; Zbl 0811.34018)] proved that the same is true for all isolated periodic minimizers, even if they are degenerate. The author proves that every (local) periodic minimizer is unstable, as a solution of the Euler-Lagrange equation, a remarkable result.
reviewed by: Alessandro Fonda
Mathematics Subject Classification ID: 34C25
zbMATH DE Number: 5323281
zbMATH Keywords: Lyapunov stability and instability
zbMATH Keywords: periodic minimizers
MaRDI profile type: Publication
full work available at URL: s00013-008-2693-x
OpenAlex ID: W2052190188
DOI: 10.1007/S00013-008-2693-X
description: scientific article; zbMATH DE number 5323281