Neumann domains of Laplace eigenfunctions

Abstract:  In this talk we introduce the Neumann domains of Laplace eigenfunctions and describe various aspects of their structure. In addition to partitioning a planar domain using the nodal domains of a Laplace eigenfunction one can also consider a natural partition formed by its gradient flow lines. Elements of such a partition are known as Neumann domains and their boundaries are called Neumann lines.  While earlier studies focused on Morse eigenfunctions (eigenfunctions with only non-degenerate critical points) many eigenfunctions exhibit more complex behavior. In this talk we discuss a framework for defining and analyzing Neumann domains even when the eigenfunction is not Morse (i.e., it has degenerate critical points)-by relying on real analyticity rather than non-degeneracy. Our approach builds on earlier works in this direction and is inspired in part by Weinberger's idea of the 'effectless cut'. We also present some asymptotics about the number of Neumann domains connecting it with the Pleijel constant.

This talk will be based on a joint work with T.V. Anoop (IIT Madras, India) and V. Bobkov (UFA Federal Research Centre, Russia).