Maximization of the second positive Neumann eigenvalue for planar domains

TL;DR

This paper establishes a sharp upper bound for the second positive Neumann eigenvalue of planar domains, proving a conjecture and providing insights into eigenvalue optimization through analytic and topological methods.

Contribution

It proves the Polya conjecture for the second Neumann eigenvalue in planar domains and introduces a novel combination of analytic and topological techniques.

Findings

The second positive Neumann eigenvalue is bounded by the first eigenvalue of a smaller disk.

The bound is sharp and achieved by degenerating domain sequences.

An upper bound for eigenvalues on conformally round metrics on spheres is also obtained.

Abstract

We prove that the second positive Neumann eigenvalue of a bounded simply-connected planar domain of a given area does not exceed the first positive Neumann eigenvalue on a disk of a twice smaller area. This estimate is sharp and attained by a sequence of domains degenerating to a union of two identical disks. In particular, this result implies the Polya conjecture for the second Neumann eigenvalue. The proof is based on a combination of analytic and topological arguments. As a by-product of our method we obtain an upper bound on the second eigenvalue for conformally round metrics on odd-dimensional spheres.