Gradient Maximum Principle for Minima

We state a maximum principle for the gradient of the minima of integral functionals

Estimates for Integral Means of Hyperbolically Convex Functions

We prove the Mejia-Pommerenke conjecture that the Taylor coefficients of hyperbolically convex functions in the disk behave like O(log^?2(n)/n) as n → ∞ assuming that the image of the unit disk under such functions is a domain of bounded boundary rotation. Moreover, we obtain some asymptotically sharp estimates for the integral means of the derivatives of such functions and consider an example of a hyperbolically convex function that maps the unit disk onto a domain of infinite boundary rotation.     

Gradient Estimates for Some Semi-Linear Hypoelliptic Equations

Given some smooth vector fields X ₁,X ₂,…,X _m on a compact manifold M, if they satisfy Hörmander’s condition, we establish global gradient estimates for the positive smooth solutions to the semi-linear hypoelliptic equations $$Lu+au\log u+bu=\partial_tu, \quad \mbox{on} \ M\times[0,\infty) $$ and $$Lu+au\log u+bu=0, \quad \mbox{on} \ M, $$ where a,b are constants, and $L=\sum_{i}X_{i}^{2}-X_{0}$ . We partially generalize the results of Cao and Yau (Math. Z. 211:485–504, 1992).     

Non-uniform Gradient Estimates for SDEs with Local Monotonicity Conditions

By using the coupling method and the localization technique, we establish non-uniform gradient estimates for Markov semigroups of diffusions or stochastic differential equations driven by pure jump Le′vy noises, where the coefficients only satisfy local monotonicity conditions.     

A Strong Maximum Principle for some quasilinear elliptic equations

In its simplest form the Strong Maximum Principle says that a nonnegative superharmonic continuous function in a domain ⁿ ,n 1, is in fact positive everywhere. Here we prove that the same conclusion is true for the weak solutions of – u + (u) = f with a nondecreasing function ,(0)=0, andf0 a.e. in if and only if the integral((s)s) ^–1/2 ds diverges ats=0+. We extend the result to more general equations, in particular to – _p u + (u) =f where _p (u) = div(|Du| ^p-2 Du), 1 <p < . Our main result characterizes the nonexistence of a dead core in some reaction-diffusion systems.This work was partly done while the author was visiting the University of Minnesota as a Fulbright Scholar.     

s-对数凸函数的Hermite-Hadamard型积分不等式

该文定义了"s-对数凸函数"的概念,并给出了可微s-对数凸函数的若干个HermiteHadamard型积分不等式,作为应用给出了平均数的几个不等式.     

Qualitative Properties of Maximum Distance Minimizers and Average Distance Minimizers in Rn

We consider one-dimensional networks of finite length in minimizing the average distance functional and the maximum distance functional subject to the length constraint. Under natural conditions, such minimizers use maximum available length, cannot contain closed loops (i.e., homeomorphic images of a circumference S¹), and have some mild regularity properties.Bibliography: 11 titles.     

Some Integral Mean Estimates for Polynomials with Restricted Zeros

Let P(z) be a polynomial of degree n having all its zeros in |z| ≤ k. Fork = 1,it is known that for each r 0 and |α|≥ 1,n(|α|- 1) {∫2π0|P(eiθ)|rdθ}1/r 0r≤ {∫2π0|1+ eiθ|rdθ}1/rmax|z|=|Dα P(z)|.In this paper, we shall first consider the case when k ≥ 1 and present certain generalizations of this inequality. Also for k ≤ 1, we shall prove an interesting result for Lacunary type of polynomials from which many results can be easily deduced.     

关于黎曼流形上的一个扩散方程的一些梯度估计

我们给出关于黎曼流形上的扩散方程θtu=Δu-▽φ·▽u（这里φ是一个C^2函数）的一些梯度估计。这推广了R.Hamilton和Qi S.Zhang关于热方程的一些梯度估计。     

A Strong Maximum Principle for parabolic equations with the p-Laplacian

We prove the Strong Maximum Principle (SMP) under suitable assumptions for a class of quasilinear parabolic problems with the p  -Laplacian, p>1$p>1$, on bounded cylindrical domains of R^N+1${\mathbb{R}}^{N+1}$,

_∂tu−_Δpu−λ|u|^p−2u≥0,${\partial }_{t}u-{\mathrm{\Delta }}_{p}u-\lambda {|u|}^{p-2}u\ge 0,$

with nonnegative initial–boundary conditions and λ≤0$\lambda \le 0$, and we give some counterexamples to the SMP if some of our assumptions are violated. We show that the Hopf Maximum Principle holds for 1<p<2$1<p<2$, and give a counterexample to it for p>2$p>2$. Also the Weak Maximum Principle for λ≤_λ1$\lambda \le {\lambda }_1$ is established.     

H型群上$p$-次Laplace算子的Hopf型引理和强极大值原理

本文首先推广了Capogna,Danielli和Garofalo关于p-次Laplace算子的径向解的一个重要公式,然后通过改进欧氏空间中证明Laplace算子的Hopf引理的方法,证明了H型群上p-次Laplace算子的Hopf型引理,进而证明了一个强极大值原理。     

On Weak and Strong Convergence of the Projected Gradient Method for Convex Optimization in Real Hilbert Spaces

This work focuses on convergence analysis of the projected gradient method for solving constrained convex minimization problems in Hilbert spaces. We show that the sequence of points generated by the method employing the Armijo line search converges weakly to a solution of the considered convex optimization problem. Weak convergence is established by assuming convexity and Gateaux differentiability of the objective function, whose Gateaux derivative is supposed to be uniformly continuous on bounded sets. Furthermore, we propose some modifications in the classical projected gradient method in order to obtain strong convergence. The new variant has the following desirable properties: the sequence of generated points is entirely contained in a ball with diameter equal to the distance between the initial point and the solution set, and the whole sequence converges strongly to the solution of the problem that lies closest to the initial iterate. Convergence analysis of both methods is presented without Lipschitz continuity assumption.     

广义线性模型极大似然估计的强相合性与渐近正态性

本文研究了若干重要类型的离散响应变量广义线性模型,在sum from i=1 to n ZiZi'的最小特征根大于cnα(对某个c>0,α>0)等条件下证明了回归参数向量的极大似然估计的强相合性与渐近正态性,其中设计阵序列{||Zn||}可以为无界序列.     

Global and Local Minimizers of the Cahn-Hilliard Functional Over a Parallelepiped: With And Without Constraint

This note is a review of results concerning the number, geometric profile and one-dimensional character of local and global minimizers of the Cahn-Hilliard functional over rectangles and parallelepipeds with and without constraint.