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Topological Methods for Variational Problems with Symmetries

  • Book
  • © 1993

Overview

Authors:
  1. Thomas Bartsch
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Part of the book series: Lecture Notes in Mathematics (LNM, volume 1560)

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Table of contents (9 chapters)

  1. Front Matter

    Pages I-X
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  2. Introduction

    • Thomas Bartsch
    Pages 1-7

  3. Category, genus and critical point theory with symmetries

    • Thomas Bartsch
    Pages 8-29

  4. Category and genus of infinite-dimensional representation spheres

    • Thomas Bartsch
    Pages 30-52

  5. The length of G-spaces

    • Thomas Bartsch
    Pages 53-71

  6. The length of representation spheres

    • Thomas Bartsch
    Pages 72-85

  7. The length and Conley index theory

    • Thomas Bartsch
    Pages 86-95

  8. The exit-length

    • Thomas Bartsch
    Pages 96-112

  9. Bifurcation for O(3)-equivariant problems

    • Thomas Bartsch
    Pages 113-126

  10. Multiple periodic solutions near equilibria of symmetric Hamiltonian systems

    • Thomas Bartsch
    Pages 127-141

  11. Back Matter

    Pages 142-154
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Keywords

About this book

Symmetry has a strong impact on the number and shape of solutions to variational problems. This has been observed, for instance, in the search for periodic solutions of Hamiltonian systems or of the nonlinear wave equation; when one is interested in elliptic equations on symmetric domains or in the corresponding semiflows; and when one is looking for "special" solutions of these problems. This book is concerned with Lusternik-Schnirelmann theory and Morse-Conley theory for group invariant functionals. These topological methods are developed in detail with new calculations of the equivariant Lusternik-Schnirelmann category and versions of the Borsuk-Ulam theorem for very general classes of symmetry groups. The Morse-Conley theory is applied to bifurcation problems, in particular to the bifurcation of steady states and hetero-clinic orbits of O(3)-symmetric flows; and to the existence of periodic solutions nearequilibria of symmetric Hamiltonian systems. Some familiarity with the usualminimax theory and basic algebraic topology is assumed.

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