Some Gruss Type Inequalities in Inner Product Spaces

不等式的中位数原理及其应用

The Median Principle for Inequalities and Applications

本文应用中位数原理推导Gruss和Ostrowski型积分不等式，提供统一证明方法。

内平行曲线的等周不等式

Isoperimetric inequalities for inner parallel curves

本文证明了光滑有界单连通区域中内平行曲线的加权等周不等式，表明固定周长下惯性矩在圆盘处最大，并将该结果推广到p≤2阶矩，同时刻画了p<3时圆盘为严格局部极大值、p>3时不等式失效的临界行为。

Riesz-type inequalities and maximum flux exchange flow

Let $D$ stand for the open unit disc in $\mathbb{R}^d$ ($d\geq 1$) and $(D,\,\mathscr{B},\,m)$ for the usual Lebesgue measure space on $D$. Let $\mathscr{H}$ stand for the real Hilbert space $L^2(D,\,m)$ with standard inner product $(\cdot,\,\cdot)$. The letter $G$ signifies the Green operator for the (non-negative) Dirichlet Laplacian $-Δ$ in $\mathscr{H}$ and $ψ$ the torsion function $G\,χ_D$. We pose the following problem. Determine the optimisers for the shape optimisation problem \[ α_t:=\sup\Big\{(Gχ_A,χ_A):\,A\subseteq D\text{is open and}(ψ,χ_A)\leq t\,\Big\} \] where the parameter $t$ …

Convergence to global equilibrium for Fokker-Planck equations on a graph and Talagrand-type inequalities

In recent work, Chow, Huang, Li and Zhou introduced the study of Fokker-Planck equations for a free energy function defined on a finite graph. When $N\ge 2$ is the number of vertices of the graph, they show that the corresponding Fokker-Planck equation is a system of $N$ nonlinear ordinary differential equations defined on a Riemannian manifold of probability distributions. The different choices for inner products on the space of probability distributions result in different Fokker-Planck equations for the same process. Each of these Fokker-Planck equations has a unique global equilibrium, whi…

二维不可压缩湍流在旋度平方-涡量-能量空间中全局吸引子的界

Bounds on the Global Attractor of 2D Incompressible Turbulence in the Palinstrophy-Enstrophy-Energy Space

针对二维不可压缩湍流全局吸引子投影解析界高估数值模拟结果的问题，指出平流项与双拉普拉斯算子内积估计不足是原因，证明Sobolev不等式上界不精确且统计各向同性湍流中该内积期望为零，讨论了其对全局吸引子行为估计的意义。