Branching in the Presence of Symmetry (CBMS-NSF Regional by David H. Sattinger

By David H. Sattinger

A dialogue of advancements within the box of bifurcation concept, with emphasis on symmetry breaking and its interrelationship with singularity idea. The notions of common strategies, symmetry breaking, and unfolding of singularities are mentioned intimately. The ebook not just studies fresh mathematical advancements but additionally offers a stimulus for extra learn within the box.

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20 CHAPTER 1 FlG. 6 Therefore the second variation of J at M* is strictly positive and W does not maximize J if the set {^ > 0} has two components. Temam (see also [13]) also considers isoperimetric problems arising from free boundary value problems. His boundary value problem is (essentially) where A and y are unspecified constants. The graph of the function / is shown in Fig. 6. Solutions of this boundary value problem are obtained as critical points of the functional on the set in the space W = {u uH 1 (H), u~ const, on F}.

18 a, b, c) above and suppose If possesses no critical values in (b, °°). Then there exist integers ko and m0 depending on b such that for all a>b, all m ^ m 0 and all R such that k0^k^m-l the homotopy group ir(BJ^a, x0) = 0 for any x0eBJ^. 17) possesses an infinite sequence of 2 IT periodic solutions {zk} where the zk are critical points of If such that If(zk)—>+&> as fc -> oo. Their results represent a remarkable improvement over the classical results in this direction, where the demonstration of even one time periodic solution was a difficult proposition.

X \ f(x) = c, /'(x) = Ax} has genus -y(Kc)^:p. The classical Ljusternik-Schnirelmann proof of this result was based on a different topological notion, that of category. 2; one simply constructs equivalent deformations. This classical result can be extended to functionals on Banach spaces which satisfy the Palais-Smale condition. For example, the functional I0(u) on the space E = {u \ \\u\\ < +°°} satisfies the Palais-Smale condition if p < (n + 2)(n -2). 19) can thus be shown to have infinitely many pairs of distinct critical points.

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