The geometry of the critical set of nonlinear periodic Sturm-Liouville operators

Burghelea, Dan; Saldanha, Nicolau C.; Tomei, Carlos

The geometry of the critical set of nonlinear periodic Sturm-Liouville operators

Authors

Burghelea, Dan

Saldanha, Nicolau C.

Tomei, Carlos

Description

We study the critical set C of the nonlinear differential operator F(u) = -u" + f(u) defined on a Sobolev space of periodic functions H^p(S^1), p >= 1. Let R^2_{xy} \subset R^3 be the plane z = 0 and, for n > 0, let cone_n be the cone x^2 + y^2 = tan^2 z, |z - 2 pi n| < pi/2; also set Sigma = R^2_{xy} U U_{n > 0} cone_n. For a generic smooth nonlinearity f: R -> R with surjective derivative, we show that there is a diffeomorphism between the pairs (H^p(S^1), C) and (R^3, Sigma) x H where H is a real separable infinite dimensional Hilbert space.
Added references, fixed typos; 24 pages, 4 figures