具间断系数拟线性椭圆型方程和方程组的正则性
本文关键词： 拟线性椭圆型方程和方程组 具VMO间断系数 A-调和逼近 自然增长条件 可控增长条件 次椭圆方程 正则性 出处：《北京交通大学》2016年博士论文 论文类型：学位论文
【摘要】：本文研究内容主要由如下四个部分组成:1、建立具VMO间断系数散度型拟线性椭圆方程组弱解的具最优Holder指数的部分Holder连续性估计;2、研究在弱条件下的具退化椭圆的A-调和型方程组弱解梯度的BMO正则性;3、得到定义在Carnot群上的具VMO间断系数的次椭圆方程组弱解梯度在Morrey空间的正则性估计;4、在自然增长条件下,分别研究半线性次椭圆方程和更一般的次椭圆A-调和方程的弱解的具最优Holder指数内部Holder连续性.下面分章节叙述具体内容:第一章简述本研究的选题背景、综述本文相关的文献资料和最新发展动态;同时也给出在正文研究中有关的基本概念和基本事实.第二章分别在可控增长条件和自然增长条件下,研究VMO间断系数的二阶散度型拟线性椭圆方程组弱解具最优Holder指数的部分Holder连续性.采用改进的A-调和逼近技术,建立方程组弱解和某个A-调和函数之间的逼近关系,再结合Caccioppoli不等式,得到在"小能量"下的Holder连续性(部分正则性).与经典的扰动法相比较,该方法避免了反向Holder不等式的使用,并在一定程度上简化了证明.第三章研究一类具弱正则系数的退化椭圆型方程组弱解梯度在全空间上的BMO正则性.基于退化椭圆型方程组弱解梯度的广义Morrey空间估计,建立了弱解梯度在BMO空间的正则性.第四章研究定义于Carnot群上在可控增长条件下具VMO系数的A-调和型次椭圆方程组,当p在2的附近扰动时其弱解梯度在Morrey空间的正则性,由此得到在Q-nλp时弱解具最优Holder指数的Holder连续性.这里需要指出的是,对于一般的p,即使是p-Laplacian,其正则性仍是未知的,文中基于反向Holder不等式,得到弱解梯度更高的可积性,通过迭代不等式,建立具确切指数的Holder连续性.第五章研究在自然增长条件下半线性次椭圆方程有界弱解的内部Holder连续性.通过线性化为线性问题的上下解问题,利用经典的De Giorgi-Moser-Nash迭代,结合向量场下的Poincare不等式和密度引理,得到Hanack不等式,从而建立方程弱解的内部Holder连续性估计.第六章考虑更一般的A-调和型次椭圆方程在自然增长条件下弱解的内部Holder连续性估计.基于密度引理和De Giorgi-Moser-Nash迭代技巧,证明A-调和型次椭圆方程的有界解的局部Holder连续性.第七章是总结和展望.
[Abstract]:This paper mainly consists of four parts: 1, and establishes the partial Holder continuity estimation with optimal Holder exponent for quasilinear elliptic equations with VMO discontinuity coefficient divergence, and studies the degenerate elliptic continuity with weak conditions. The BMO regularity of the gradient of weak solutions for A- harmonic equations is obtained. The regularity estimates for the gradient of weak solutions of subelliptic equations with VMO discontinuity coefficients defined on Carnot groups in Morrey space are obtained. 4. Under the condition of natural growth, The interior Holder continuity of semi-linear sub-elliptic equation and more general sub-elliptic A-harmonic equation with optimal Holder exponent is studied respectively. This paper summarizes the relevant literature and the latest developments, and also gives the basic concepts and basic facts in the main body research. Chapter two, under the conditions of controllable growth and natural growth, respectively. In this paper, the partial Holder continuity of the weak solutions of second order divergence type quasilinear elliptic equations with VMO discontinuity coefficients with optimal Holder exponent is studied. By using the improved Aharmonic approximation technique, the approximation relations between the weak solutions of the equations and some A- harmonic functions are established. Combined with Caccioppoli inequality, the Holder continuity (partial regularity) under "small energy" is obtained. Compared with the classical perturbation method, this method avoids the use of reverse Holder inequality. In chapter 3, we study the BMO regularity of the gradient of weak solutions of a class of degenerate elliptic systems with weak regular coefficients in the whole space. Based on the generalized Morrey space estimation of the gradient of weak solutions of degenerate elliptic equations, The regularity of weak solution gradient in BMO space is established in chapter 4th. In chapter 4th, we study the regularity of the gradient of weak solution in Morrey space when p is perturbed near 2 under the condition of controllable growth on Carnot group. The Holder continuity of the weak solution with the optimal Holder exponent is obtained at Q-n 位 p. It should be pointed out here that the regularity of the weak solution is unknown for the general p, even p-Laplacian. Based on the reverse Holder inequality, the higher integrability of the weak solution gradient is obtained. The Holder continuity with exact exponents is established by iterative inequalities. Chapter 5th studies the internal Holder continuity of bounded weak solutions of semilinear subelliptic equations under natural growth conditions. By using the classical de Giorgi-Moser-Nash iteration, Poincare inequality and density Lemma under vector field, the Hanack inequality is obtained. In chapter 6th, we consider the interior Holder continuity estimation of the weaker solutions of the A- harmonic subelliptic equation under the condition of natural growth. Based on the density Lemma and de Giorgi-Moser-Nash iterative technique, The local Holder continuity of bounded solutions for subelliptic equations of A-harmonic type is proved. Chapter 7th is a summary and prospect.
【学位授予单位】：北京交通大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175.25
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