The geometry of the critical set of nonlinear periodic Sturm-Liouville operators
نویسندگان
چکیده
We study the critical set C of the nonlinear differential operator F (u) = −u′′+f(u) defined on a Sobolev space of periodic functions Hp(S1), p ≥ 1. Let Rxy ⊂ R3 be the plane z = 0 and, for n > 0, let ⊲⊳n be the cone x2 + y2 = tan2 z, |z − 2πn| < π/2; also set Σ = Rxy ∪ ⋃ n>0 ⊲⊳n. For a generic smooth nonlinearity f : R → R with surjective derivative, we show that there is a diffeomorphism between the pairs (Hp(S1), C) and (R3,Σ)×H where H is a real separable infinite dimensional Hilbert space.
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