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arXiv · 1209.6469

An optimal Poincaré-Wirtinger inequality in Gauss space

Barbara Brandolini·Francesco Chiacchio·Antoine Henrot·Cristina Trombetti
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Abstract

Let $Ω$ be a smooth, convex, unbounded domain of $\R^N$. Denote by $μ_1(Ω)$ the first nontrivial Neumann eigenvalue of the Hermite operator in $Ω$; we prove that $μ_1(Ω) \ge 1$. The result is sharp since equality sign is achieved when $Ω$ 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 $H^1(Ω,dγ_N)$, where $γ_N$ is the $N$-dimensional Gaussian measure.

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Barbara Brandolini, Francesco Chiacchio, Antoine Henrot, Cristina Trombetti. 2012-10-05. An optimal Poincaré-Wirtinger inequality in Gauss space. https://arxiv.org/abs/1209.6469

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