An optimal Poincar\'e-Wirtinger inequality in Gauss space

TL;DR

This paper establishes an optimal Poincaré-Wirtinger inequality in Gaussian space, proving that the first nontrivial Neumann eigenvalue of the Hermite operator in convex domains is at least 1, with equality in strip domains.

Contribution

The paper proves a sharp lower bound for the first nontrivial Neumann eigenvalue of the Hermite operator in convex domains, establishing an optimal inequality in Gaussian weighted Sobolev spaces.

Findings

The first nontrivial Neumann eigenvalue in convex domains is at least 1.

Equality is achieved in strip-shaped domains.

The result provides an optimal inequality in Gaussian space.

Abstract

Let $\Omega$ be a smooth, convex, unbounded domain of $\R^N$. Denote by $\mu_1(\Omega)$ the first nontrivial Neumann eigenvalue of the Hermite operator in $\Omega$; we prove that $\mu_1(\Omega) \ge 1$. The result is sharp since equality sign is achieved when $\Omega$ is a $N$-dimensional strip. Our estimate can be equivalently viewed as an optimal Poincar\'e-Wirtinger inequality for functions belonging to the weighted Sobolev space $H^1(\Omega,d\gamma_N)$, where $\gamma_N$ is the $N$-dimensional Gaussian measure.