The geometry of the critical set of nonlinear periodic Sturm–Liouville operators (original) (raw)
2009, Journal of Differential Equations
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 ⊂ R 3 be the plane z = 0 and, for n > 0, let ⊲⊳ n be the cone x 2 + y 2 = tan 2 z, |z − 2πn| < π/2; also set Σ = R 2 xy ∪ n>0 ⊲⊳ 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 , Σ) × H where H is a real separable infinite dimensional Hilbert space.
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