An optimal Poincaré-Wirtinger inequality in Gauss space

2012/09/28 by Brandolini, Barbara, Chiacchio, Francesco, Henrot, Antoine +1
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1209.6469

Abstract

Let Ω be a smooth, convex, unbounded domain of \R^N. Denote by μ₁(Ω) the first nontrivial Neumann eigenvalue of the Hermite operator in Ω; we prove that μ₁(Ω) ≥ 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 H¹(Ω,dγ_N), where γ_N is the N-dimensional Gaussian measure.

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