Closed curves in \((\mathbb R^3)\) with prescribed curvature and torsion in perturbative cases. I: Necessary condition and study of the unperturbed problem
type: Item
instance of: scholarly article
title: Closed curves in \((\mathbb R^3)\) with prescribed curvature and torsion in perturbative cases. I: Necessary condition and study of the unperturbed problem
zbMATH Open document ID: 1229.53004
DOI: 10.4171/RLM/466
author: Paolo Caldiroli
author: Michela Guida
published in: Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Serie IX. Rendiconti Lincei. Matematica e Applicazioni
publication date: 2007-08-07T00:00:00Z
review text: Summary: We study the problem of \((\kappa, \tau)\)-loops, i.e. closed curves in the three-dimensional Euclidean space, with prescribed curvature \(\kappa\) and torsion \(\tau\). We state a necessary condition for the existence of a bounded sequence of \((\kappa_{n}, \tau_{n})\)-loops when the functions \(\kappa_n\) and \(\tau_n\) converge to the constants 1 and 0, respectively. Moreover we prove some Fredholm-type properties for the ``unperturbed'' problem, with \(\kappa \equiv 1\) and \(\tau \equiv 0\).
Mathematics Subject Classification ID: 53A04
Mathematics Subject Classification ID: 47A53
zbMATH DE Number: 5177498
zbMATH Keywords: prescribed curvature and torsion
zbMATH Keywords: perturbative methods
zbMATH Keywords: Fredholm operators
MaRDI profile type: Publication
cites work: Perturbation of \(\Delta u+u^{(N+2)/(N-2)}=0\), the scalar curvature problem in \(\mathbb{R}^N\), and related topics
cites work: A multiplicity result for the Yamabe problem on \(S^n\)
cites work: Perturbation methods and semilinear elliptic problems on \(\mathbb R^n\)
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description: scientific article; zbMATH DE number 5177498