The equality case in a Poincaré-Wirtinger type inequality

Abstract

In this paper, generalizing to the non smooth case already existing results, we prove 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. Furthermore, and this is our main result, under some additional assumptions on $\Omega$, 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.

References in corpus (1)

• A sharp lower bound for some Neumann eigenvalues of the Hermite operator

Cited by in corpus (2)

• The Friedland-Hayman inequality and Caffarelli's contraction theorem
• Optimal Szegö-Weinberger type inequalities