The equality case in a Poincaré–Wirtinger type inequality

Abstract

It is known that, for any convex planar set $\Omega$, the first non-trivial Neumann eigenvalue ${\mu }_1(\Omega )$ of the Hermite operator is greater than or equal to 1. Under the additional assumption that $\Omega$ is contained in a strip, we show that ${\mu }_1(\Omega )=1$ if and only if $\Omega$ is any strip. The study of the equality case requires, among other things, an asymptotic analysis of the eigenvalues of the Hermite operator in thin domains.