能量不等式

Secondly, through integrating by parts and using Cauchy inequality, thanks to a lemma of Gronwalls type, an energy inequality is got and a priori estimate is established.

接下来经过分部积分和Cauchy不等式，并利用Gronwall型引理得到一个能量不等式，对解的范数给出先验估计。

Secondly, through integrating by parts and using Cauchy inequality, thanks to a lemma of Gronwall's type, an energy inequality is got and a priori estimate is established.

接下来经过分部积分和Cauchy不等式，并利用Gronwall型引理得到一个能量不等式，对解的范数给出先验估计。

First, we establish an energy inequality in the case that no boundary conditions are imposed, and obtain a uniqueness result in the Cauchy problem for the model.

首先，对没有任何边界条件的方程给出一个能量不等式，并且获得Koiter壳的一个Cauchy问题的唯一性结果。

In the case of Dirichlet boundary conditions,these estimates were improved by Baker[1]who used a technique that can be interpreted as a nonstandard energy argument.

Dupont[15]使用标准能量方法给出了一类线性双曲方程Galerkin解的L2误差估计，对于Dirichlet型边值问题，Baker[1]对其结果作了改进，用的是一种可谓"非标准的能量不等式"方法。

The paper first obtains the L2 -apriori estimates for the solutions of two kinds of autocatalytic model s under Dirichlet boundary conditions, and then, using the properties of linear semigroup and delicate calculations, the estimates of the maximal norms are obtained, therefore, the global existence of the solutions is proved.

该文利用重要不等式及能量积分方法首先得到了解的初等的先验估计。然后利用线性半群的有关性质及精细计算得到了解的最大模估计，从而证明了两类三次自催化模型在Dirichlet边界条件下整体解的存在性，并进而证明了第一类模型的最大吸引子的存在性

The main aim of this thesis is to study the properties of the maps for Heisenberg group target, which include Lipschitz and Holder continuity, L~#,H~(n , W~(1,p)#,H~(n , BMO and John-Nirenberg estimates, embedded theorems, Poincare inequalities and reverse Poincare inequalities, the regul-arities about the minimizers.

本文的主要目的是系统研究靶流形为Heisenberg群的函数及其空间的性质，其中包括Lipschitz及Hlder连续性、空间L~p及W~(1，p)的性质、空间BMO的性质及其上的John-Nirenberg估计、嵌入定理、Poincare不等式和逆Poincare不等式、能量极小映射的存在性、正则性及用调和函数逼近能量极小映射等问题。

In this paper we will mainly study the gauge fixing Yang-Mills heat flow of a principal bundle For a principle bundle with a compact seme-simple Lie group as its structure group over a compact Riemannian manifold without boundary, the evolutions of the curvature and its higher derivatives under the flow above will be derived, and the energy inequality and the Bochner type estimates will be obtained.

中文摘要：本论文主要讨论主丛上的规范固定Yang-Mills热流我们在紧致无边黎曼流形上的以半单紧致李群为结构群的主丛上，推导了在规范固定Yang-Mills热流下曲率及其高阶导数的演化方程，得到了该热流的能量不等式和Bochner估计，由此可推出单调性公式和小作用量正则性，以及曲率各阶导数的局部一致估计。

In this paper we will mainly study the gauge fixing Yang-Mills heat flow of a principal bundleFor a principle bundle with a compact seme-simple Lie group as its structure group over a compact Riemannian manifold without boundary, the evolutions of the curvature and its higher derivatives under the flow above will be derived, and the energy inequality and the Bochner type estimates will be obtained. Then, the monotonicity formula and the small action regularity theorem can be proved. We will give the locally uniform estimates for the higher derivatives of the curvature.

本论文主要讨论主丛上的规范固定Yang-Mills热流我们在紧致无边黎曼流形上的以半单紧致李群为结构群的主丛上，推导了在规范固定Yang-Mills热流下曲率及其高阶导数的演化方程，得到了该热流的能量不等式和Bochner估计，由此可推出单调性公式和小作用量正则性，以及曲率各阶导数的局部一致估计。

The energy inequality and parabolic monotonici ty inequality is essential for the regularity of a weak solution to this problem .

该问题弱解的能量不等式和抛物单调不等式在证明弱解的正则性时起关键作用。

The thesis consists of three parts:The first part deals with optimal Sobolev inequalities on compact Riemannian manifolds;the second part is concerned with some variational models in image processing,and we propose variational models as well as experimental results for detection,denoising and mosaicking;the last part deals with inverse differential geometry in 3-D non-local image restoration.

本论文由三个部分组成。第一部分是流形上的非线性分析：第二部分是基于能量极小化方法的图像处理；第三部分是三维非局部图像重建及逆向微分几何。在第一部分中，我们研究了紧致流形上的最优Sobolev不等式，证明了某些不等式虽然就整体而言是不成立的，但总是局部成立的。

The absorption and distribution of chromium were studied in ryeusing nutrient culture technique and pot experiment.

采用不同浓度K2CrO4(0，0.4，0.8和1.2 mmol/L)的Hoagland营养液处理黑麦幼苗，测定铬在黑麦体内的亚细胞分布、铬化学形态及不同部位的积累。

By analyzing theory foundation of mathematical morphology in the digital image processing, researching morphology arithmetic of the binary Image, discussing two basic forms for the least structure element: dilation and erosion.

通过分析数学形态学在图像中的理论基础，研究二值图像的形态分析算法，探讨最小结构元素的两种基本形态：膨胀和腐蚀；分析了数学形态学复杂算法的基本原理，把数学形态学的部分并行处理理念引入到家实际应用中。