Infinite Dimensional Isoperimetric Inequalities in Product Spaces with the Supremum Distance

We study the isoperimetric problem for product probability measures with respect to the uniform enlargement. We construct several examples of measures for which the isoperimetric function of coincides with the one of the infinite product . This completes earlier works by Bobkov and Houdré.     

$p$-双调和算子的第一特征值的等周上界和 Reilly 型不等式

在本文中,我们给出了嵌入到欧氏空间中的$n$维闭超曲面上$p$-双调和算子的第一特征值的一些等周上界.我们也给出了浸入到高维流形如欧氏空间,球面和射影空间中的闭子流形上$p$-双调和算子的第一特征值的一些Reilly-型不等式.     

The sharp Sobolev inequality and the Banchoff-Pohl inequality on surfaces

Let be a complete two dimensional simply connected Riemannian manifold with Gaussian curvature . If is a compactly supported function of bounded variation on , then satisfies the Sobolev inequality

Conversely, letting be the characteristic function of a domain recovers the sharp form of the isoperimetric inequality for simply connected surfaces with . Therefore this is the Sobolev inequality ``equivalent' to the isoperimetric inequality for this class of surfaces. This is a special case of a result that gives the equivalence of more general isoperimetric inequalities and Sobolev inequalities on surfaces.

Under the same assumptions on , if is a closed curve and is the winding number of about , then the Sobolev inequality implies

which is an extension of the Banchoff-Pohl inequality to simply connected surfaces with curvature .

     

Logarithmic Sobolev Inequalities,Matrix Models and Free Entropy

We give two applications of logarithmic Sobolev inequalities to matrix models and free probability. We also provide a new characterization of semi-circular systems through a Poincare-type inequality.     

Mass Transport and Variants of the Logarithmic Sobolev Inequality

We develop the optimal transportation approach to modified log-Sobolev inequalities and to isoperimetric inequalities. Various sufficient conditions for such inequalities are given. Some of them are new even in the classical log-Sobolev case. The idea behind many of these conditions is that measures with a non-convex potential may enjoy such functional inequalities provided they have a strong integrability property that balances the lack of convexity. In addition, several known criteria are recovered in a simple unified way by transportation methods and generalized to the Riemannian setting. The research of A.V. Kolesnikov was supported by RFBR 07-01-00536, DFG Grant 436 RUS 113/343/0 and GFEN 06-01-39003.     

Hilbert不等式的各种精化与拓广综述

在本文中,我们较全面的概述了1990年至2002年间有关Hilbert级数型与积分型不等式的种种精化与拓广工作.涉及发表于海内外的文献资料40余篇,其中我国学者的工作成果占有重要份量.文末还列举了一些有待进一步研究的公开问题.     

Sharp Sobolev Embeddings and Related Hardy Inequalities: The Sub -- Critical Case

The paper deals with sharp embeddings of the spaces B and F into rearrangement-variant spaces and related Hardy inequalities. Here (1/_p, s) belongs to the interior of the shaded invariant spaces region in the Figure     

矩阵的特征值和特征向量同时求解的一种方法

本文给出了通过λ－矩阵的初等变换，同时求得特征值和特征向量的一种方法。     

生灭过程与一维扩散过程的对数sobolev不等式

本文运用加权的Hardy不等式的方法给出了生灭过程与一维扩散过程满足对数Sobolev不等式的显式判别准则。     

Recovery of Immersions from Their Metric Tensors and Nonlinear Korn Inequalities: A Brief Survey

The authors discuss the existence and uniqueness up to isometries of Enof immersions φ : Ω  R~n→ E~n with prescribed metric tensor field(g ij) : Ω→ S~n, and discuss the continuity of the mapping(gij) →φ defined in this fashion with respect to various topologies. In particular, the case where the function spaces have little regularity is considered. How, in some cases, the continuity of the mapping(gij) →φ can be obtained by means of nonlinear Korn inequalities is shown.     

解Stokes特征值问题的一种两水平稳定化有限元方法

基于局部Gauss积分,研究了解Stokes特征值问题的一种两水平稳定化有限元方法．该方法涉及在网格步长为H的粗网格上解一个Stokes特征值问题,在网格步长为h=O(H2)的细网格上解一个Stokes问题．这样使其能够仍旧保持最优的逼近精度,求得的解和一般的稳定化有限元解具有相同的收敛阶,即直接在网格步长为h的细网格上解一个Stokes特征值问题．因此,该方法能够节省大量的计算时间．数值试验验证了理论结果．     

Pseudomonotone Variational Inequalities: Convergence of the Auxiliary Problem Method

This paper deals with the convergence of the algorithm built on the auxiliary problem principle for solving pseudomonotone (in the sense of Karamardian) variational inequalities.     

一种求非负不可约三对角矩阵最大特征值的方法

就非负不可约三对角矩阵,给出了一种求最大特征值的方法,关键是求迭代因子g的新方法,且证明了此迭代因子大于文献[2]中的迭代因子(r+3d)/(r+2d),从而减少了迭代次数,节约了运算时间.     

The Cayley Method and the Inverse Eigenvalue Problem for Toeplitz Matrices

Despite the fact that symmetric Toeplitz matrices can have arbitrary eigenvalues, the numerical construction of such a matrix having prescribed eigenvalues remains to be a challenge. A two-step method using the continuation idea is proposed in this paper. The first step constructs a centro-symmetric Jacobi matrix with the prescribed eigenvalues in finitely many steps. The second step uses the Cayley transform to integrate flows in the linear subspace of skew-symmetric and centro-symmetric matrices. No special geometric integrators are needed. The convergence analysis is illustrated for the case of n = 3. Numerical examples are presented.     

The Brascamp–Lieb Inequalities: Finiteness, Structure and Extremals

We consider the Brascamp–Lieb inequalities concerning multilinear integrals of products of functions in several dimensions. We give a complete treatment of the issues of finiteness of the constant, and of the existence and uniqueness of centred gaussian extremals. For arbitrary extremals we completely address the issue of existence, and partly address the issue of uniqueness. We also analyse the inequalities from a structural perspective. Our main tool is a monotonicity formula for positive solutions to heat equations in linear and multilinear settings, which was first used in this type of setting by Carlen, Lieb, and Loss [CLL]. In that paper, the heat flow method was used to obtain the rank-one case of Lieb’s fundamental theorem concerning exhaustion by gaussians; we extend the technique to the higher-rank case, giving two new proofs of the general-rank case of Lieb’s theorem. Received: September 2005 Revision: November 2005 Accepted: November 2005     

Polynomials,Higher Order Sobolev Extension Theorems and Interpolation Inequalities on Weighted Folland-Stein Spaces on Stratified Groups

This paper consists of three main parts. One of them is to develop local and global Sobolev interpolation inequalities of any higher order for the nonisotropic Sobolev spaces on stratified nilpotent Lie groups. Despite the extensive research after Jerison's work [3] on Poincaré-type inequalities for Hörmander's vector fields over the years, our results given here even in the nonweighted case appear to be new. Such interpolation inequalities have crucial applications to subelliptic or parabolic PDE's involving vector fields. The main tools to prove such inqualities are approximating the Sobolev functions by polynomials associated with the left invariant vector fields on ?. Some very usefull properties for polynomials associated with the functions are given here and they appear to have independent interests in their own rights. Finding the existence of such polynomials is the second main part of this paper. Main results of these two parts have been announced in the author's paper in Mathematical Research Letters [38].The third main part of this paper contains extension theorems on anisotropic Sobolev spaces on stratified groups and their applications to proving Sobolev interpolation inequalities on (?,δ) domains. Some results of weighted Sobolev spaces are also given here. We construct a linear extension operator which is bounded on different Sobolev spaces simultaneously. In particular, we are able to construct a bounded linear extension operator such that the derivatives of the extended function can be controlled by the same order of derivatives of the given Sobolev functions. Theorems are stated and proved for weighted anisotropic Sobolev spaces on stratified groups.     

Nonconvex energy functions,related eigenvalue hemivariational inequalities on the sphere and applications

The present paper deals with a new type of eigenvalue problems arising in problems involving nonconvex nonsmooth energy functions. They lead to the search of critical points (e.g. local minima) for nonconvex nonsmooth potential functions which in turn give rise to hemivariational inequalities. For this type of variational expressions the eigenvalue problem is studied here concerning the existence and multiplicity of solutions by applying a critical point theory appropriate for nonsmooth nonconvex functionals.     

乘幂法求矩阵特征向量与特征值的初始向量及循环控制

矩阵迭代法是求矩阵的第一阶特征值与特征向量的一种数值方法 .本文讨论了用矩阵迭代法求解矩阵的特征值与特征向量时的初始向量选取和循环控制条件     

关于Jacobi矩阵的特征值反问题可解的充分必要条件的一个注记 (英)

本文讨论一类具有特殊结构的Jacobi矩阵的特征值反问题,该问题由描述变截面杆的微分方程离散化得到．我们得到了这个问题有解的一些必要条件,并且通过一些数值例子,说明了L．Lu和K．Michael给出的充分条件和算法在矩阵的阶数高于3的时候是错误的。     

Dirichlet Forms,Poincaré Inequalities,and the Sobolev Spaces of Korevaar and Schoen

We answer a question of Jost on the validity of Poincaré inequalities for metric space-valued functions in a Dirichlet domain. We also investigate the relationship between Dirichlet domains and the Sobolev-type spaces introduced by Korevaar and Schoen.