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

TL;DR

This paper characterizes the geometric structure of the critical set of a nonlinear Sturm-Liouville operator with periodic boundary conditions, revealing a diffeomorphic relation to a specific geometric cone structure.

Contribution

It provides a detailed geometric description of the critical set for nonlinear periodic Sturm-Liouville operators, establishing a diffeomorphism to a cone-based geometric model.

Findings

Critical set is diffeomorphic to a geometric cone structure.

The geometric model involves a union of a plane and cones in three-dimensional space.

The result applies to generic smooth nonlinearities with surjective derivatives.

Abstract

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.