Bartsch T. — Topological Methods for Variational Problems with Symmetries :: Электронная библиотека попечительского совета мехмата МГУ

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Bartsch T. — Topological Methods for Variational Problems with Symmetries

Bartsch T. — Topological Methods for Variational Problems with Symmetries

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Название: Topological Methods for Variational Problems with Symmetries

Автор: Bartsch T.

Аннотация:

Lecture Notes in Mathematics, 1560

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.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1993

Количество страниц: 156

Добавлена в каталог: 07.05.2010

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Предметный указатель

$\mathcal{A}$ -category      12 31f
$\mathcal{A}$ -category, properties      15—17 19
$\mathcal{A}$ -genus      12f 31f
$\mathcal{A}$ -genus, properties      17—19
Admissible representations      24 34 48
Antisymplectic matrix      128
Attractor      87
Attractor-repeller pair      87
B-index      13f
bg      56
Borel cohomology      55f 82
Borel stable cohomotopy      80
Borsuk — Ulam function      49f
Borsuk — Ulam theorem      33 52 75
Brake orbit      138f
Burnside ring      39 81 83
Category, equivariant      12
Center manifold theorem      109f
Classifying space      56
CN-group      37
Coefficient ring      54
Cohomological index      61f
Compact equivariant category      23
Conjugacy class      9
Conley index      88
Connecting orbit      87 91 120 123
Connection matrix      93
Continuation      88 94 99
Continuity property of A-category      15
Continuity property of A-genus      18
Continuity property of cohomology theory      54f
Continuity property of exit-length      99
Continuity property of length      63
Continuous cohomology theory      54
Cup-length      57
Cup-product      54
Cyclic group      46f 50 76f
Deformation monotonicity      15
Degree function      36
Dirichlet problem      27f
EG      55
eigenvalue problems      21 23f
Entry length      97
Equivariant cohomology theory      54 (see also Borel cohomology)
Equivariant cohomology theory, K-theory      77
Equivariant cohomology theory, stable cohomotopy      80 83
Equivariant cup-length      58
Euler class      74 79
Excision property of the length      63
Exit length      97—99
Exit length, properties      99 101f
Exit set      88
Fixed point set      9
Flow      87
Flow, gradient-like      87
Flow, parametrized      88
Frobeiiius group      37
G-jargon, ANR      11
G-jargon, Banach space      10
G-jargon, CW-complex      10
G-jargon, Hilbert space      10
G-jargon, map      9
G-jargon, Morse decomposition      87
G-jargon, orbit      9
G-jargon, space      9
G-jargon, vector space, normed      10
Generalized symplectic matrix      128
Genus      12f
Genus, properties      17—19
Geometrical index      14
Gradient-like flow      87
Grassmann manifold      56
Groups, 3-step group      37
Groups, CN-group      37
Groups, cyclic group      46f 50 76f
Groups, Frobenius group      37
Groups, O(3)      114—117
Groups, p-group      31 48—50 80
Groups, p-toral group      31 33f
Groups, p-torus      22 73
Groups, semisimple Lie group      32f
Groups, simple group      32f
Groups, SO(3)      47 114
Groups, torus      22 73
Gysin sequence      74
Hamiltonian system      128
Homogeneous G-space      9

Hopf bifurcation      141
Hyperbolic      93f
Index pair      88
Induced representation      35
Intersection property of the length      63
Invariant sphere theorem      111
Invariant subset      87
Isolated invariant set      87
Isolating neighborhood      87
Isotropy group      9
Join      12
K-theory, equivariant      77
Lattice of isotropy subgroups      116
Length      59
Length for $S^{1}$       60
Length for $\mathbb{Z}/p$       60f 67
Length, properties      63—65 68
Limit sets $(\alpha-,\omega-)$       87
Lyapunov function      87
Maximal invariant subset      87
Monotonicity property of $\mathcal{A}$ -genus      17
Monotonicity property of length      63
Morse decomposition      87
Mountain Pass Theorem      21
Multiplicative cohomology theory      54
Noetherian module      63
Noetherian ring      57
Normal mode solution      139f
Normalization property of $\mathcal{A}$ -category      15
Normalization property of $\mathcal{A}$ -genus      17
Normalization property of length      63 68
Normed G-vector space      10
Orbit      9
Orbit space      9
Orthogonal group O(3), lattice of isotropy subgroups      116
Orthogonal group O(3), representations      115
Orthogonal group O(3), subgroups      114f
p-group      31 48—50 80
p-toral group      31 33f
p-toral Sylow subgroup      32
p-torus      22 73
Palais — Smale condition      20
Parametrized flow      90f
Piercing property of length      65
Positively invariant      88
Principles of symmetry reduction      31
Product flow      88
Properties of $\mathcal{A}$ -category      15—17 19
Properties of $\mathcal{A}$ -genus      17—19
Properties of cohomology theories      54
Properties of length      63—65 68
Regular index pair      88
Related by continuation      89
Repeller      87
Representation of O(3)      115
Representation of SO(3)      47
Representation, admissible      24 34 48
Representation, induced      35
Segal conjecture      81
Semisimple Lie group      32f
Set valued genus      14
Signature      134
Simple group      32f
Space of conjugacy classes      9f 35
Special orthogonal group SO(3), irreducible representations      47
Special orthogonal group SO(3), subgroups      47 114
Spherical harmonics      47 115
Stable cohomotopy theory      80 83
Stable set      97
Stiefel manifold      56
Strong normalization property      68
Subadditivity property of $\mathcal{A}$ -category      15
Subadditivity property of $\mathcal{A}$ -genus      18
Subadditivity property of length      63
Subconjugate      10
Submaximal isotropy group      118
Symmetry breaking      118
Symplectic matrix      128
Telescope      42f
Tietze — Gleason theorem      10
Torus group      22 73
Triangle inequality for length      63
Unstable set      97
Weak monotonicity property      19
Weinstein — Moser theorem      129 138f
Weyl group      9
Whitehead theorem      11

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