On energy functionals, Kähler-Einstein metrics, and the Moser-Trudinger-Onofri neighborhood

We prove that the existence of a Kähler-Einstein metric on a Fano manifold is equivalent to the properness of the energy functionals defined by Bando, Chen, Ding, Mabuchi and Tian on the set of Kähler metrics with positive Ricci curvature. We also prove that these energy functionals are bounded from below on this set if and only if one of them is. This answers two questions raised by X.-X. Chen. As an application, we obtain a new proof of the classical Moser-Trudinger-Onofri inequality on the two-sphere, as well as describe a canonical enlargement of the space of Kähler potentials on which this inequality holds on higher-dimensional Fano Kähler-Einstein manifolds.     

The Gaussian Isoperimetric Inequality and Transportation

Any probability measure on ^d which satisfies the Gaussian or exponential isoperimetric inequality fulfils a transportation inequality for a suitable cost function. Suppose that W (x) dx satisfies the Gaussian isoperimetric inequality: then a probability density function f with respect to W (x) dx has finite entropy, provided that t . This strengthens the quadratic logarithmic Sobolev inequality of Gross (Amr. J. Math 97 (1975) 1061). Let (dx) = e ^–(x) dx be a probability measure on ^d, where is uniformly convex. Talagrand's technique extends to monotone rearrangements in several dimensions (Talagrand, Geometric Funct. Anal. 6 (1996) 587), yielding a direct proof that satisfies a quadratic transportation inequality. The class of probability measures that satisfy a quadratic transportation inequality is stable under multiplication by logarithmically bounded Lipschitz densities.     

Talagrand's T_2-Transportation Inequality and Log-Sobolev Inequality for Dissipative SPDEs and Applications to Reaction-Diffusion Equations

We establish Talagrand's T2-transportation inequalities for infinite dimensional dissipative diffusions with sharp constants, through Galerkin type's approximations and the known results in the finite dimensional case. Furthermore in the additive noise case we prove also logarithmic Sobolev inequalities with sharp constants. Applications to Reaction-Diffusion equations are provided.     

Hardy's Inequality for Dirichlet Forms

We prove an abstract form of Hardy's L² inequality, in which the Dirichlet integral is replaced by the Dirichlet form of a general symmetric Markov process. A number of examples are provided.     

投影下的Gronwall不等式

本文对Ｊ．Ｋ．Ｈａｌｅ曾提出的一类广泛的投影下的Ｇｒｏｎｗａｌ不等式问题作了讨论，对满足ｕ（ｔ）≤ａ（ｔ）＋∫_０^tｂ（ｔ－ｓ）ｕ（ｓ）ｄｓ＋∫_０^∞ｃ（ｓ）ｕ（ｔ＋ｓ）ｄｓ，(?)ｔ≥０的函数ｕ（ｔ）∈Ｃ_b⁰（Ｒ₊，Ｒ₊）作了估计．其结果对讨论微分方程的有界解、不变流形及其Ｆｏｌｉａｔｉｏｎ和进一步讨论奇性Ｇｒｏｎｗａｌ不等式都有意义     

Talagrand＇s T2-Transportation Inequality and Log-Sobolev Inequality for Dissipative SPDEs and Applications to Reaction-Diffusion Equations

Abstract We establish Talagrand’s T₂-transportation inequalities for infinite dimensional dissipative diffusions with sharp constants, through Galerkin type’s approximations and the known results in the finite dimensional case. Furthermore in the additive noise case we prove also logarithmic Sobolev inequalities with sharp constants. Applications to Reaction-Diffusion equations are provided. * Project supported by the Yangtze Scholarship Program.     

A General Duality Scheme for Nonconvex Minimization Problems with a Strict Inequality Constraint

We establish a duality formula for the problem Minimize f(x)+g(x) for h(x)+k(x)<0 where g, k are extended-real-valued convex functions and f, h belong to the class of functions that can be written as the lower envelope of an arbitrary family of convex functions. Applications in d.c. and Lipschitzian optimization are given.     

A-调和方程弱解的双权Caccioppoli型不等式

研究形如div A(x,u(x))=0的A-调和方程,证明了其弱解满足局部Aλr双权Caccioppoli型不等式.其中算子A:Ω×Rn→Rn满足如下条件:对于正常数0   

索伯列夫嵌入不等式的证明

We consider the problem about the space embedded by tile space and the embed-ding inequality. With the Holder inequality and interpolation inequality, we give the proof of the space embedding theorem and the space holder embedding theorem.     

Biharmonic Equation and an Improved Hardy Inequality

An improved Hardy inequality will be proven in the present work.Using the improved Hardy in-equality and variational techniques,we also discuss the existence of nontrivial solution for following the weightedeigenvalue problem:(?)     

一类积分不等式及其应用

研究了一类新的非线性延迟积分不等式φ(u(t))≤(n_1(t)+∫_o~(a(t)) f_1(s)w_1(u(s))ds)(n_2(t)+∫_o~t f_2(s)w_2)(u9(s))ds),t∈R_+得到了新的结论,推广了已有的若干结果.并举例说明其应用.     

关于Shafer-Fink不等式和Carlson不等式

改进了Shafer-Fink不等式的上界,并给出了Carlson不等式的一个上界估计.     

Jensen's Inequality for Backward Stochastic Differential Equations

Under the Lipschitz assumption and square integrable assumption on g, the author proves that Jensen's inequality holds for backward stochastic differential equations with generator g if and only if g is independent of y, g(t, 0) = 0 and g is super homogeneous with respect to z. This result generalizes the known results on Jensen's inequality for g-expectation in [4, 7-9].     

Hlder不等式和Minkowski不等式的推广

本文利用变分方法对多个变元的不含变元导数的Holder不等式和Minkowski不等式进行了推广．此种方法的主要意义不在于证明传统的不等式，而在于发现新的不等式．     

条件g-期望的矩不等式

找到了几个使条件g-期望的矩不等式在一般意义下成立的关于g和g-期望的充分条件.     

参数变分不等式的灵敏性分析

本文在所给函数和映射均不可微的前提下，通过建立参数变分不等式和参数Wiener-Hopf方程的等价性，分析了Hilbert空间中参数变分不等式的局部唯一解的灵敏性。文中所用方法是N.D.Yen之方法的改进，使用这一方法可大大简化N.D.Yen一文中主要结果（引理2.1)的证明。     

可积情形下的Gronwall不等式

在[6]讨论J．K．Hale指出的投影下Gronwall不等式u(t)≤ a(t)＋∫₀^tb(t-s)u(s)ds＋∫₀^∞c(s)u(t＋s)ds,?t≥ 0,基础上．指明[6]所要求的(?)a(t)=0是可以去掉的．进而,本文去掉了u(t),a(t)和c(t)的连续性以及a(t)的单调性,仅在可积性条件下得到了更一般的结论．所得的结果不仅真正包含了[5]的结果,而且新的证明方法使[5]中的疏漏得以补正．     

The FKG Inequality for Partially Ordered Algebras

The FKG inequality asserts that for a distributive lattice with log-supermodular probability measure, any two increasing functions are positively correlated. In this paper we extend this result to functions with values in partially ordered algebras, such as algebras of matrices and polynomials. This research was supported by an NSF grant.     

一个不等式的证明

用两种办法给出了沐定夷,谢惠民在其《吉米多维奇数学分析习题集学习指引》第二册中提出的一个未解决的不等式的证明．     

反应扩散方程解的渐近性态

研究了一类反应扩散方程初始边值问题. 利用微分不等式理论得到了问题解的渐近性态.