Mathematical Research Letters

Volume 20 (2013)

Number 3

An optimal Poincaré-Wirtinger inequality in Gauss space

Pages: 449 – 457

DOI: http://dx.doi.org/10.4310/MRL.2013.v20.n3.a3

Authors

Barbara Brandolini (Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli “Federico II”, Italy)

Francesco Chiacchio (Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli “Federico II”, Italy)

Antoine Henrot (Institut Elie Cartan, Université de Lorraine, Vandoeuvreles-Nancy, France)

Cristina Trombetti (Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli “Federico II”, Italy)

Abstract

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 $\mu_1(\Omega) \geq 1$. The result is sharp since equality sign is achieved when $\Omega$ is an $N$-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.

Keywords

Neumann eigenvalue, Hermite operator, sharp bounds

2010 Mathematics Subject Classification

35B45, 35J70, 35P15

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