An isoperimetric problem with two distinct solutions

2022/10/31 by Henrot, Antoine, Lemenant, Antoine, Lucardesi, Ilaria
#52A10 #52A40 #Analysis of PDEs (math.AP) #FOS: Mathematics #Primary 35P15 Secondary: 49Q10 #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2210.17225

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 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.

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