AI 中文总结
该研究针对满足特定条件的连通Lipschitz区域，证明了Ornstein-Uhlenbeck算子Neumann特征值的尖锐高斯调和平均不等式，确定等号成立的充要条件，结合Ritz论证与高斯重排完成证明。
AI 中文摘要
设$N\geq2$，$\Omega\subset\mathbb{R}^N$是连通Lipschitz区域，可能无界，关于原点对称且满足$0<\gamma_N(\Omega)<1$。假设高斯Sobolev嵌入$H^1(\Omega,\gamma_N)\hookrightarrow L^2(\Omega,\gamma_N)$是紧的，充分条件是存在从$\Omega$到$\mathbb{R}^N$的有界高斯Sobolev延拓算子。记正Ornstein-Uhlenbeck算子$-\Delta+x\cdot\nabla$在$\Omega$中的Neumann特征值为$0=\mu_0(\Omega)<\mu_1(\Omega)\leq\mu_2(\Omega)\leq\cdots$。我们证明尖锐倒数和不等式$\sum_{k=1}^{N}\frac{1}{\mu_k(\Omega)} \geq \frac{N}{\mu_1(B_R)}$，其中$B_R$是中心在原点的欧氏球，满足$\gamma_N(B_R)=\gamma_N(\Omega)$，等号当且仅当$\Omega=B_R$。证明结合耦合N维Ritz论证与高斯射线重排，角失衡由对称无迹矩阵编码，其贡献由有限维凸性不等式控制。
英文摘要
Let $N\geq2$ and let $Ω\subset\R^N$ be a connected Lipschitz domain, possibly unbounded, symmetric with respect to the origin, and such that $0<\gammaN(Ω)<1$. We assume that the Gaussian Sobolev embedding $H^1(Ω,\gammaN)\hookrightarrow L^2(Ω,\gammaN)$ is compact; a sufficient condition is the existence of a bounded Gaussian Sobolev extension operator from $Ω$ to $\R^N$. Denote by \[ 0=μ_0(Ω)<μ_1(Ω)\leqμ_2(Ω)\leq\cdots \] the Neumann eigenvalues of the positive Ornstein--Uhlenbeck operator $-Δ+x\cdot\nabla$ in $Ω$. We prove the sharp reciprocal-sum inequality \[ \sum_{k=1}^{N}\frac{1}{μ_k(Ω)} \geq \frac{N}{μ_1(B_R)}, \] where $B_R$ is the Euclidean ball centred at the origin and satisfying $\gammaN(B_R)=\gammaN(Ω)$. Equality holds if and only if $Ω=B_R$. The proof combines a coupled $N$-dimensional Ritz argument with a Gaussian raywise rearrangement. The angular imbalance is encoded by a symmetric trace-free matrix, whose contribution is controlled by a finite-dimensional convexity inequality.
