上有界的

In first chapter,we discuss the〓-integrability of thegradients of the very weak solutions to non-linear elliptic systemsdivA=B(2)and〓-regularity of the very weak or the weak solutions to non-linearelliptic systems〓(3)in bounded domain Ω.

第一章讨论有界区域上的非线性椭圆型方程组〓（2）的很弱解的梯度的〓可积性和非线性椭圆型方程组〓（3）的很弱解与弱解的〓正则性。

Thirdly, we will concentrate on the multiplicity of the positive and minimal sign-changing solutions of the following quasilinear equations with Hardy-Sobolev critical exponent in bounded domain By Lusternik-Schnirelmann category theory, we showthe relation the number of positive solutions with the topology of the domain.

再次研究有界区域上具有Hardy-Sobolev临界指数的拟线性方程的正解与极小变号解的多重性，运用Lusternik-Schnirelmann理论，给出方程正解与区域的拓扑性质的相关性。

Based on general orthogonal polynomials and general hybrid orthogonal functions, a new linear, continuous and bounded operator has been proposed in the dissertation. The definitions and properties of PGOPO are given, a various operational rules of the operator have been set up in a systematic way, the convergence of approximate solutions has been discussed. After introducing PGOPO and discussing its main properties and operational rules, a systematic methodology for solving various control problems via PGOPO has been developed.

本文在广义正交多项式及广义混合正交函数的基础上，提出了一种新的线性、连续、有界算子—分段广义正交多项式算子，探讨了这种算子的基本性质，并系统地建立了与该算子有关的各种运算规则及收敛性结论，给出了它在控制理论中的各种应用。

In order to effectively abstract the texture and denoise but not damage the edge and other important details of image, Meyer (2001) proposed using elements of BV space and G space to characterize cartoon and details of image respectively. The authors present PDE of minimizing energy on the basis of Meyer's work, After the model is discretizated. projection algorithm and the method for ROF model are applied. Thus an image is decomposed into two parts, bounded variation component u and texture with noise υ.

为有效地提取纹理和去噪而不损坏图像的边缘及其他重要细节，Meyer(2001)提出了分别用BV空间和G空间刻画图像的主体和细节部分，笔者在Meyer的图像分解模型基础上，建立能量最小化PDE方程，将模型离散化后，利用投影算法和ROF模型的求解方法，将图像分解为有界变差部分u和包含纹理和噪声的部分υ。

Furthermore, we define a convolution multiplication between characteristic functions of constructible subsets by using push-forward functor from the category of algebraic varieties over C to the category of spaces of constructible functions. We construct geometric model for "intrinsic symmetry" of the octahedral axiom in a triangulated category. Using it, we deduce the multiplication satisfies the Jacobi identity of Lie algebra and then realize infinite dimensional Lie algebras.

进一步，我们使用复代数簇范畴到可构函数空间范畴的pushforward函子，给出了可构集上特征函数的卷积乘法，并构造了三角范畴八面体公理的内蕴对称性的几何模型，最终证明了对于不可分解支撑有界可构集的特征函数，乘法满足李代数定义的Jacobi恒等式，从而给出了无限维李代数的实现。

Applying the convexification of nonconvex function and the theorem in Chapter 2, we establish the perturbed optimization in the bounded subsets of the dual of locally convex spaces

运用非凸函数凸化的方法以及第二章给出的命题，在局部凸空间的对偶空间的有界子集上建立相应的的扰动优化和变分原理

Using the method of blow up lower and upper solutions, the existence and boundary behavior of positive boundary blow up solutions for some elliptic systems of competitive type with singular weight functions are obtained, here the singular weight functions are permitted to be bounded in some parts of the boundary, and to go to infinity or even oscillate in other parts.

利用爆破上、下解方法，本文建立了带奇异权函数的竞争型椭圆方程组正的边界爆破解的存在性和边界行为，这里奇异权函数可以在边界的某一部分有界，而在其他的部分趋于无穷或者震荡。

In order to effectively Abstract the texture and denoise but not damage the edge and other important details of image,Meyer(2001) proposed using elements of BV space and G space to characterize cartoon and details of image respectively.The authors present PDE of minimizing energy on the basis of Meyer s work.

为有效地提取纹理和去噪而不损坏图像的边缘及其他重要细节，Meyer(2001)提出了分别用BV空间和G空间刻画图像的主体和细节部分，笔者在Meyer的图像分解模型基础上，建立能量最小化PDE方程，将模型离散化后，利用投影算法和ROF模型的求解方法，将图像分解为有界变差部分u和包含纹理和噪声的部分v。

The main results and novelties of the paper are as follows:Firstly, we study the existence of nontrivial solution for the following nonlinear equations with singular pontential By using variational methods and critical point theory, we construct linking type critical value of this kind of singular elliptic equations, with the properties of the eigenvalue, we estimate the critical value and prove the local condition holds to show the existence of the nontrivial solution.

论文的创新点及主要结果如下：首先研究有界区域上具有奇异位势的非线性方程的非平凡解的存在性，运用变分方法与临界点理论，构造这一类奇异椭圆方程的环绕型临界值，并由特征值的性质给出临界值的估计，同时证明局部条件成立，最终证明方程非平凡解的存在性。

The stars of this site are two rabbits who patiently provided the poses I've used to illustrate the text: Betsy and Marvin (a tricolored, broken-patterned mix). They're supermodels of the rabbit world.

大意：本文当中充当模特任我蹂躏的两个兔明星叫做Betsy 和Marvin (那只身上有三种颜色的混种兔，下文简称M)，真是兔界的超模啊。

Singer Leona Lewis and former Led Zeppelin guitarist Jimmy Page emerged as the bus transformed into a grass-covered carnival float, and the pair combined for a rendition of "Whole Lotta Love".

歌手leona刘易斯和前率领的飞艇的吉他手吉米页出现巴士转化为基层所涵盖的嘉年华花车，和一双合并为一移交&整个lotta爱&。

This is Kate, and that's Erin.

这是凯特，那个是爱朗。