An isoperimetric problem with two distinct solutions

TL;DR

This paper identifies the convex domains with two axes of symmetry that maximize the first non-trivial Neumann eigenvalue under perimeter constraints, showing that the square and equilateral triangle are optimal, and introduces a new bound involving minimal width.

Contribution

It proves that the square and equilateral triangle maximize the Neumann eigenvalue among symmetric convex domains, and introduces a new bound involving minimal width and area.

Findings

Square and equilateral triangle are maximizers.

New bound on rmf3n eigenvalue involving minimal width.

Partially answers a question from 2009.

Abstract

In this paper we prove that among all convex domains of the plane with two axis of symmetry, the maximizer of the first non trivial Neumann eigenvalue $\mu_1$ with perimeter constraint is achieved by the square and the equilateral triangle. Part of the result follows from a new general bound on $\mu_1$ involving the minimal width over the area. Our main result partially answers to a question addressed in 2009 by R. S. Laugesen, I. Polterovich, and B. A. Siudeja.