An optimal Poincaré-Wirtinger inequality in Gauss space

Barbara Brandolini (1) , Francesco Chiacchio (1) , Antoine Henrot (2) , Cristina Trombetti (1)

1 Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”
2 IECL - Institut Élie Cartan de Lorraine

Let $\Omega$ be a smooth, convex, unbounded domain of $\mathbb{R}^N$. 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 Poincaré-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.

Mots clés: Neumann eigenvalue; Hermite operator; sharp bounds

Domaines: Equations aux dérivées partielles [math.AP]

Barbara Brandolini, Francesco Chiacchio, Antoine Henrot, Cristina Trombetti. An optimal Poincaré-Wirtinger inequality in Gauss space. Mathematical Research Letters, 2013, 20 (3), pp.449-457. ⟨hal-00814754⟩