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TRANSFORMATION GROUPS AND BIFURCATION AT MULTIPLE EIGENVALUES 1

By D. H. Sattinger, Communicated Hans and F. Weinberger

Abstract

1. Topological degree arguments show that bifurcation must take place at eigenvalues of odd multiplicity, while examples show bifurcation may not take place at eigenvalues of even multiplicity. The general problem of bifurcation at multiple eigenvalues is one which does not readily submit to a complete solution, so the approach must proceed by special cases. One important class for which a more detailed analysis is possible is that of problems invariant under a transformation group. The purpose of this note is to present some results in this direction. We would also like to call the reader's attention to the recent interesting work of D. Ruelle. 2. Consider a nonlinear problem (1) (L0 + XLJu + N(u) = 0 where L0,L1 and N are continuous operators from a Banach space X to 7(1 c Y). We assume that N is Fréchet differentiable, that JV'(0) = 0, and that L0 + XLY is a Fredholm operator for all X. (This means that the dimension of the null space of L0 + XLX is equal to the codimension of its range in Y, and that its range is closed in Y.) Let G be the group generated by S 1 (the unit circle with the usual addition) together with an inversion operator i(i 2 = e9 where e is the identity operator). Let Tg be a representation of G on Y. We label elements of S 1 by y9 <5, where 0 ^ y, ô < In. Then TôTy = Tô+r We assume that L0 + XLX and N commute with Tg for all g in G. Let jVk be the null space of (L0 + ALJ and let {Xj} be the real numbers X for which ylf is nontrivial. J ^ is invariant under G, and we assume J£j is finite dimensional. If Jfx. is irreducible, it is either one or two dimensional. Since bifurcation at simple eigenvalues is well understood (see [5] or [3]), let us consider the case dim Jfx. = 2. The eigenvalues of Tt are ± 1, so in J£. there is one vector q>j such that T^- = cpj. Restricting (1) to the subspaces X ' and Y ' of all vectors u such that u — Ttu, the corresponding null space J ^ is one dimensional. (Since N commutes with Tf, TtN(u) = N{Ttu) = N(u) if u is in X +.) We can now apply the usual bifurcation arguments to obtain a bifurcating curve of solutions AMS (MOS)subject classifications (1970). Primary 35J60, 20C35. Key words and phrases. Transformation groups, bifurcation theory

Year: 2013
OAI identifier: oai:CiteSeerX.psu:10.1.1.306.3651
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