The topology of critical sets of some ordinary differential operators

Nicolau C. Saldanha, Carlos Tomei

Published 2005-01-05Version 1

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.