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
modified by lucardesi on 02 Oct 2023


Accepted Paper

Inserted: 1 nov 2022
Last Updated: 2 oct 2023

Journal: Transactions of the American Mathematical Society
Year: 2023

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.