Applications of degree for \(S^1\)-equivariant gradient maps to variational nonlinear problems with \(S^1\)-symmetries
type: Item
instance of: scholarly article
title: Applications of degree for \(S^1\)-equivariant gradient maps to variational nonlinear problems with \(S^1\)-symmetries
zbMATH Open document ID: 0891.55003
DOI: 10.12775/TMNA.1997.018
author: Sławomir Maciej Rybicki
published in: Topological Methods in Nonlinear Analysis
publication date: 1998-07-12T00:00:00Z
review text: The author [Nonlinear Anal., Theory Methods Appl. 23, No. 1, 83-102 (1994; Zbl 0815.58027)] established a degree theory for \(S^1\)-equivariant orthogonal maps. If \(V\) is a finite-dimensional orthogonal representation of \(S^1\), call \(D_\alpha(V):=\{x\in V||x|<\alpha\}\) for \(\alpha>0\). Let \(f:V\to\mathbb{R}\) be a \(C^2\)-function such that \(0\in V\) is an isolated critical point and choose \(\alpha>0\) such that \((\text{grad}f)^{-1}(\{0\})\cap D_\alpha(V)=\{0\}\). Then the degree \(\text{DEG}(\text{grad}f,D_\alpha(V))\) is defined as an element in \(\mathbb{Z}\oplus\bigoplus_{n=1}^\infty\mathbb{Z}\) . On the one hand, this degree is a powerful tool in an equivariant situation; on the other hand, the definition is fairly involved and computing the degree is extremely complicated. The author's objective in this article is twofold: First, he computes the equivariant degree for \(S^1\)-equivariant gradient maps, in particular product maps, in a variety of special situations. He then applies these results to obtain sufficient conditions for the existence of bifurcation points of \(S^1\)-equivariant variational bifurcation problems. In a similar way, the author deduces sufficient criteria implying the existence of nontrivial orbits of zeros of \(S^1\)-equivariant gradient maps.
Mathematics Subject Classification ID: 55M25
Mathematics Subject Classification ID: 58E09
zbMATH DE Number: 1130754
zbMATH Keywords: degree
zbMATH Keywords: equivariant map
zbMATH Keywords: variational problem
zbMATH Keywords: bifurcation
reviewed by: Christian Fenske
MaRDI profile type: Publication
full work available at URL: tmna.1997.018
OpenAlex ID: W2579684730
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description: scientific article; zbMATH DE number 1130754