Counting symmetry-breaking solutions to symmetric variational problems
The author develops a geometric theory of the action of a compact Lie group \(G\) on a finite-dimensional smooth manifold \(M\). For equivariant dynamical systems \(\dot x = f(x)\) \((f(gx)+\) \(dg \cdot f)(x))\) in variational case these results allow to apply the Ljusternik-Schnirelman category for obtaining the lower bound on the number of solutions with given symmetry \(H \subseteq G\). Concerning the ``equivariant branching lemma'', see also the works [Sov. Math., Dokl. 20, 586-590 (1979); translation from Dokl. Akad. Nauk SSSR 246, 1048-1051 (1979; Zbl 0429.47028)] and [``Branching theory of solutions of nonlinear equations under group invariance conditions'', Tashkent, ``Fan'' (1985; Zbl 0593.58028)] by the reviewer.
scientific article; zbMATH DE number 859293
scientific article