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Topological methods for variational problems with symmetries. (English) Zbl 0789.58001

Lecture Notes in Mathematics. 1560. Berlin: Springer-Verlag. x, 152 p. (1993).
It is well known that symmetry has a strong effect on the number and shape of solutions to variational problems. As the starting point of a critical point theory for symmetric functionals may be considered a theorem of Lyusternik and Schnirelman: An even \(C^ 2\)-map \(S^ n\to \mathbb{R}\) must have at least \(n+1\) pairs of antipodal critical points. This follows by computing the Lyusternik-Schnirelman category \(\text{cat}(\mathbb{R} P^ n)\) of the orbit space \(\mathbb{R} P^ n\). The equality \(\text{cat}(\mathbb{R} P^ n) = n+1\) is a consequence of the Borsuk- Ulam theorem (which states that there does not exist an odd map \(S^ n \to S^{n-1}\)). A natural question arises here: What can we say if the symmetry group \(\mathbb{Z}/2\) in the above theorem is replaced by more general groups? In most papers in this field the considered groups are the cyclic groups \(\mathbb{Z}/p\) of prime order and the group \(S^ 1\) of complex numbers of modulus 1. For more general groups, restrictive assumptions are needed to reduce the general results to the corresponding \(\mathbb{Z}/p\)-version or \(S^ 1\)-version. The question is: what version of the Borsuk-Ulam theorem one needs to generalize a result from \(\mathbb{Z}/2\) to other symmetry groups? Here is one of the motivations of this book.
The author presents recent techniques and results in critical point theory for functionals invariant under a symmetry group. In the present book we find both main methods in this field: Lyusternik-Schnirelman category and Morse theory. First, a version of Lyusternik-Schnirelman category in spaces with group actions is given. A corresponding notion of genus is also defined. Some basic properties are followed by a critical point theory for functionals with symmetries. In order to obtain an effective computation of category and genus, a new cohomological index theory for spaces with group actions is developed. The corresponding index is called length and is an equivariant version of the cup-length. The length is used to obtain lower bounds for category and genus. It is computed in a number of cases (not only for \(\mathbb{Z}/2\) or \(S^ 1\)). As a consequence, a generalization of the Borsuk-Ulam theorem is obtained. Finally, the length in combination with the Morse-Conley index theory is used to study gradient-like flows with symmetries. The author considers the length of the unstable set of an isolated invariant set of a flow and applies it to bifurcation problems. He shows that a change of this length along a branch of stationary solutions gives rise to bifurcation. The theory is illustrated with two applications: the bifurcation of steady states and heteroclinic orbits of \(O(3)\)-symmetric flow, and the existence of periodic solutions near equilibria of symmetric Hamiltonian systems.
The reading of the present book is very stimulating. The reader will find here many new (and well motivated) ideas which are integrated in a clear and elegant exposition. Certainly, this work will be a starting point for further research.
Reviewer: C.Popa (Iaşi)

MSC:

58-02 Research exposition (monographs, survey articles) pertaining to global analysis
58Exx Variational problems in infinite-dimensional spaces
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
55P91 Equivariant homotopy theory in algebraic topology
58E40 Variational aspects of group actions in infinite-dimensional spaces
58E07 Variational problems in abstract bifurcation theory in infinite-dimensional spaces
55N25 Homology with local coefficients, equivariant cohomology
49R50 Variational methods for eigenvalues of operators (MSC2000)
34Cxx Qualitative theory for ordinary differential equations
35J20 Variational methods for second-order elliptic equations
37Cxx Smooth dynamical systems: general theory
55M30 Lyusternik-Shnirel’man category of a space, topological complexity à la Farber, topological robotics (topological aspects)
47J05 Equations involving nonlinear operators (general)
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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