Topological Aspects of Critical Points and Level Sets in Elliptic PDEs

Abstract

In this paper we discuss the emergence of topological structures through the critical points and level sets of solutions to elliptic PDEs. First, we analyze the local and global properties of the critical points of Green’s functions on complete Riemannian manifolds, showing that the number of critical points on a surface of finite type admits a topological upper bound, a property that does not hold in higher dimensions. Second, we introduce two technical tools, Thom’s isotopy theorem and a Runge-type global approximation theorem, which will allow us to construct solutions to a wide range of elliptic PDEs with level sets of complicated (sometimes bizarre) topologies. The model elliptic equation that we consider to illustrate the power of these techniques is the Helmholtz equation (monochromatic waves). These ideas are used in two seemingly unrelated applications: a 2001 conjecture of Sir Michael Berry about the existence of Schrodinger operators in Euclidean space with eigenfunctions having nodal lines of arbitrary knot type, and the construction of bounded solutions to the Allen-Cahn equation with level sets of any compact topology. The aim of these notes is not to provide detailed proofs of all the stated results but to introduce the main ideas and methods behind certain selected topics in the study of topological structures in PDEs. This is the set of lecture notes the second named author gave at the CIME Summer School held in Cetraro (Italy) during June 2017.