Calculus of Variations and Geometric Measure Theory

A. Henrot - A. Lemenant - I. Lucardesi

An isoperimetric problem with two distinct solutions

created by lemenant on 01 Nov 2022



Inserted: 1 nov 2022
Last Updated: 1 nov 2022

Year: 2022

ArXiv: 2210.17225 PDF


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 μ1 with perimeter constraint is achieved by the square and the equilateral triangle. Part of the result follows from a new general bound on μ1 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.