INFINITELY MANY SOLUTIONS FOR A CLASS OF DEGENERATE ELLIPTIC EQUATIONS

Let X = （X1, ···, Xm） be an infinitely degenerate system of vector fields. The aim of this paper is to study the existence of infinitely many solutions for the sum of operators X =sum （ ） form j=1 to m Xj Xj.

Existence and regularity of solutions to semi-linear Dirichlet problem of infinitely degenerate elliptic operators with singular potential term

In this paper,we study the Dirichlet problem for a class of semi-linear infinitely degenerate elliptic equations with singular potential term.By using the logarithmic Sobolev inequality and Hardy's inequality,the existence and regularity of multiple nontrivial solutions have been proved.

A De Giorgi Type Result to Divergence Degenerate Elliptic Equation with Bounded Coefficients Related to Hörmander's Vector Fields

In this paper,we consider the divergence degenerate elliptic equation with bounded coefficients constructed by Hörmander's vector fields.We prove a De Giorgi type result,i.e,the local Holder continuity for the weak solutions to the equation by providing a De Giorgi type lemma and extending the Moser iteration to the setting here.As a consequence,the Harnack inequality of weak solutions is also given.

THE EXPONENTIAL PROPERTY OF SOLUTIONS BOUNDED FROM BELOW TO DEGENERATE EQUATIONS IN UNBOUNDED DOMAINS

This paper is focused on studying the structure of solutions bounded from below to degenerate elliptic equations with Neumann and Dirichlet boundary conditions in unbounded domains.After establishing the weak maximum principles,the global boundary Holder estimates and the boundary Harnack inequalities of the equations,we show that all solutions bounded from below are linear combinations of two special solutions(exponential growth at one end and exponential decay at the other)with a bounded solution to the degenerate equations.

DECAY RATE FOR DEGENERATE CONVECTION DIFFUSION EQUATIONS IN BOTH ONE AND SEVERAL SPACE DIMENSIONS

We consider degenerate convection-diffusion equations in both one space dimension and several space dimensions. In the first part of this article, we are concerned with the decay rate of solutions of one dimension convection diffusion equation. On the other hand, in the second part of this article, we are concerned with a decay rate of derivatives of solution of convection diffusion equation in several space dimensions.

C^(α)regularity of weak solutions of non-homogeneous ultraparabolic equations with drift terms

We consider a class of non-homogeneous ultraparabolic differential equations with singular drift terms arising from some physical models,and prove that weak solutions are H?lder continuous,which are sharp in some sense and also generalize the well-known De Giorgi-Nash-Moser theory to degenerate parabolic equations satisfying the H?rmander hypoellipticity condition.The new ingredients are manifested in two aspects:on the one hand,for lower-order terms,we exploit a new Sobolev inequality suitable for the Moser iteration by improving the result of Pascucci and Polidoro(2004);on the other hand,we explore the G-function from an early idea of Kruzhkov(1964)and an approximate weak Poincaréinequality for non-negative weak sub-solutions to prove the H?lder regularity.

REGULARITY OF SOLUTIONS TO AN ELLIPTIC VARIATIONAL INEQUALITY SYSTEM

Even for elliptic variational inequality systems with degenerate ellipticity in the form of (1) the boundedness and regularity are unsolved for general obstacle problems. In this paper the CI’cr regularity of solutions is proved only for a special case of obstacle problems of (1).

The regularity of the surface of the =Gauss curvature 0≤ K∈C_0~∞

Under the mild conditions, it is proved that the convex su rf ace is global C 1,1 , with the given Gaussian curvature 0≤K∈C ∞ 0 a nd the given boundary curve. Examples are given to show that the regularity is o ptimal.

Smoothness of the Gradient of Weak Solutions of Degenerate Linear Equations

Let Q（x） be a nonnegative definite, symmetric matrix such that √Q（X） is Lipschitz con- tinuous. Given a real-valued function b（x） and a weak solution u（x） of div（QVu） = b, we find sufficient conditions in order that √Qu has some first order smoothness. Specifically, if is a bounded open set in Rn, we study when the components of vVu belong to the first order Sobolev space W1＇2（Ω） defined by Sawyer and Wheeden. Alternately we study when each of n first order Lipschitz vector field derivatives Xiu has some first order smoothness if u is a weak solution in Ω of ^-^-1 X^Xiu ＋ b = O. We do not assume that {Xi}is a HSrmander collection of vector fields in ~. The results signal ones for more general equations.

Configurations of Shock Regular Reflection by Straight Wedges

We are concerned with the shock regular reflection configurations of unsteady global solutions for a plane shock hitting a symmetric straight wedge.It has been known that patterns of the shock reflection are various and complicated,including the regular and the Mach reflection.Most of the fundamental issues for the shock reflection have not been understood.Recently,there are great progress on the mathematical theory of the shock regular reflection problem,especially for the global existence,uniqueness,and structural stability of solutions.In this paper,we show that there are two more possible configurations of the shock regular reflection besides known four configurations.We also give a brief proof of the global existence of solutions.

Exponential Growth Solutions of Elliptic Equations

We show that for a class of second order divergence form elliptic equations on an infinite strip with the Dirichlet boundary condition,the space of fixed order exponential growth solutions is of finite dimension.An optimal estimation of the dimension is given.Examples also show that the finiteness property may not be true if one drops some of the conditions we make in our result.

一类退缩的椭圆型方程弱解的正则性

本文给出了一类退缩的拟线性椭圆型方程－ Ｄｉｖ［｜ Δｕ｜ｐ－ ２ Δｕ ＋ Ｆ（ｘ，ｕ）］ ＝ Ｂ（ｘ，ｕ， Δｕ）在Ｗ１，ｐ（Ω）中弱解的Ｃ１，λｌｏｃ （Ω）正则性，其中Ω为ＲＮ 中任一区域．

一类退化半线性椭圆方程的Pontryagin最大值原理

研究一类由退化半线性椭圆方程所支配的分布参数系统的最优控制问题。由于系统的退化性,当退化点集为单点集时,利用正则化方法得到正则化方程解的一些一致估计,再通过一个逼近过程证明了非退化系统最优控制的存在性。利用变分思想,得到该分布参数系统的Pontryagin最大值原理。

一类散度型椭圆方程的很弱解

本文研究满足一致椭圆型条件的散度型椭圆方程 ∑ni,j=1ddxjai,j(x) dudxi =0的很弱解 ,并以Hodge分解和弱逆H lder不等式为工具 ,证明了其正则性结果 :对任意的 2 - 2 n+ 1× 1 0 0 n2βα( 2 n+ 2 + 1 ) 2 -1<r≤ 2 ,都存在可积指数p =p n ,βα,r >2 ,使得对其任意很弱解u∈W1r,loc(Ω) ,都有u∈W1p ,loc(Ω) .特别 ,u是其通常意义下的弱解 .

一类多调和方程的可解性研究

多调和方程问题的研究是椭圆形偏微分方程边值问题研究的热点之一,文章通过将多调和方程边值问题转换成椭圆形方程组问题,利用不动点原理以及上调和函数的极值原理,证明了多调和方程边值问题正解的存在性;同时,证明了一定条件下正解的惟一性,最后讨论了正解的不存在性。

一类退化半线性椭圆方程支配系统的最优控制条件

研究了一类由退化半线性椭圆方程所支配的分布参数系统的最优控制问题.当退化点集的测度为零时,利用正则化方法和变分思想,得到了该分布参数系统最优控制所满足的必要条件.

一类二阶退化半线性椭圆型方程边值问题的适定性及解的正则性

本文考虑一类二阶退化半线性椭圆型方程边值问题.由椭圆正则化方法建立能量不等式,利用紧性推理,Banach—Saks定理,弱解与强解一致性,解常微分方程,椭圆型方程正则性定理,迭代方法.极值原理和Fredholm—Riesz-Schauder理论,可得相应线性问题适定性及解的高阶正则性;再由Moser引理和Banach不动点定理可得半线性问题解的存在性.这类问题与几何中无穷小等距形变刚性问题密切相关,其高阶正则性解的存在性对几何应用尤为重要.

一类退化椭圆型方程Dirichlet问题的解的高阶正则性

由于退化椭圆型方程的研究与双曲空间中极小图的Dirichlet问题,以及曲面的无穷小等距形变刚性问题的密切联系,在有界周期域上讨论了一类退化椭圆型方程Dirichlet问题的解的高阶正则性,利用泛函分析方法得到一个涉及解的高阶正则性的充分必要条件.

一类退化椭圆型方程弱解的性质

研究了一类退化椭圆型方程.当退化点集的测度为零时,利用正则化方法和Sobolev嵌入定理证明了该方程弱解的存在性和唯一性.

蜕化椭圆型变分不等式解的C^(1,α)正则性

讨论一类蜕化椭圆型变分不等式解的 C1 ,α正则性 ,并解决了 L indgvist及