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.
Boxing inequalities for relative fractional perimeter and fractional Poincaré-type inequalities on John domains with the BBM factor
Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), School of Mathematics and Statistics, Hunan Normal University（湖南师范大学数学与统计学院计算与随机数学教育部重点实验室）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
Log-Sobolev and Beckner inequalities and stability of Poincaré inequality with weighted Gaussian measures
School of Science and the Environment, Grenfell Campus, Memorial University of Newfoundland（纪念大学格伦菲分校科学与环境学院）；Department of Mathematics, University of Connecticut（康涅狄格大学数学系）
Stability of the $L^{p}$-Poincaré inequality for the Lebesgue measure and Gaussian probability measure with explicit geometric dependence and applications to spectral gaps
KIMEP University（基梅普大学）；SDU University（SDU大学）；Institute of Mathematics and Mathematical Modeling, Kazakhstan（哈萨克斯坦数学与数学建模研究所）
Sharp $L^2$-stability and gradient stability hierarchies of the $L^2$-Poincaré inequalities on Euclidean balls and Gaussian Poincaré inequality
School of Science and the Environment, Grenfell Campus, Memorial University of Newfoundland（纽芬兰纪念大学格伦菲尔夫学院科学与环境学院）；Department of Mathematics, University of Connecticut（康涅狄格大学数学系）
