The topology of critical sets of some ordinary differential operators

Details The topology of critical sets of some ordinary differential operators The topology of critical sets of some ordinary differential operators

2005-01-05

arxiv.org/abs/math/0501071v1

Progress in Nonlinear Differential Equations and Their Applications, Vol. 66, 491-504, 2005.

Mathematics

Functional Analysis

15 pages, 4 figures

Scientific paper

We survey recent work of Burghelea, Malta and both authors on the topology of critical sets of nonlinear ordinary differential operators. For a generic nonlinearity $f$, the critical set of the first order nonlinear operator $F_1(u)(t) = u'(t) + f(u(t))$ acting on the Sobolev space $H^1_p$ of periodic functions is either empty or ambient diffeomorphic to a hyperplane. For the second order operator $F_2(u)(t) = -u''(t) + f(u(t))$ on $H^2_D$ (Dirichlet boundary conditions), the critical set is ambient diffeomorphic to a union of isolated parallel hyperplanes. For second order operators on $H^2_p$, the critical set is not a Hilbert manifold but is still contractible and admits a normal form. The third order case is topologically far more complicated.

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