本文關(guān)鍵詞： 擬線性橢圓型方程和方程組 具VMO間斷系數(shù) A-調(diào)和逼近 自然增長條件 可控增長條件 次橢圓方程 正則性　出處：《北京交通大學(xué)》2016年博士論文　論文類型：學(xué)位論文

【摘要】：本文研究內(nèi)容主要由如下四個(gè)部分組成:1、建立具VMO間斷系數(shù)散度型擬線性橢圓方程組弱解的具最優(yōu)Holder指數(shù)的部分Holder連續(xù)性估計(jì);2、研究在弱條件下的具退化橢圓的A-調(diào)和型方程組弱解梯度的BMO正則性;3、得到定義在Carnot群上的具VMO間斷系數(shù)的次橢圓方程組弱解梯度在Morrey空間的正則性估計(jì);4、在自然增長條件下,分別研究半線性次橢圓方程和更一般的次橢圓A-調(diào)和方程的弱解的具最優(yōu)Holder指數(shù)內(nèi)部Holder連續(xù)性.下面分章節(jié)敘述具體內(nèi)容:第一章簡述本研究的選題背景、綜述本文相關(guān)的文獻(xiàn)資料和最新發(fā)展動(dòng)態(tài);同時(shí)也給出在正文研究中有關(guān)的基本概念和基本事實(shí).第二章分別在可控增長條件和自然增長條件下,研究VMO間斷系數(shù)的二階散度型擬線性橢圓方程組弱解具最優(yōu)Holder指數(shù)的部分Holder連續(xù)性.采用改進(jìn)的A-調(diào)和逼近技術(shù),建立方程組弱解和某個(gè)A-調(diào)和函數(shù)之間的逼近關(guān)系,再結(jié)合Caccioppoli不等式,得到在"小能量"下的Holder連續(xù)性(部分正則性).與經(jīng)典的擾動(dòng)法相比較,該方法避免了反向Holder不等式的使用,并在一定程度上簡化了證明.第三章研究一類具弱正則系數(shù)的退化橢圓型方程組弱解梯度在全空間上的BMO正則性.基于退化橢圓型方程組弱解梯度的廣義Morrey空間估計(jì),建立了弱解梯度在BMO空間的正則性.第四章研究定義于Carnot群上在可控增長條件下具VMO系數(shù)的A-調(diào)和型次橢圓方程組,當(dāng)p在2的附近擾動(dòng)時(shí)其弱解梯度在Morrey空間的正則性,由此得到在Q-nλp時(shí)弱解具最優(yōu)Holder指數(shù)的Holder連續(xù)性.這里需要指出的是,對(duì)于一般的p,即使是p-Laplacian,其正則性仍是未知的,文中基于反向Holder不等式,得到弱解梯度更高的可積性,通過迭代不等式,建立具確切指數(shù)的Holder連續(xù)性.第五章研究在自然增長條件下半線性次橢圓方程有界弱解的內(nèi)部Holder連續(xù)性.通過線性化為線性問題的上下解問題,利用經(jīng)典的De Giorgi-Moser-Nash迭代,結(jié)合向量場下的Poincare不等式和密度引理,得到Hanack不等式,從而建立方程弱解的內(nèi)部Holder連續(xù)性估計(jì).第六章考慮更一般的A-調(diào)和型次橢圓方程在自然增長條件下弱解的內(nèi)部Holder連續(xù)性估計(jì).基于密度引理和De Giorgi-Moser-Nash迭代技巧,證明A-調(diào)和型次橢圓方程的有界解的局部Holder連續(xù)性.第七章是總結(jié)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.
【學(xué)位授予單位】：北京交通大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類號(hào)】：O175.25

【參考文獻(xiàn)】

相關(guān)期刊論文 前8條

1 邱亞林;;REGULARITY FOR NONLINEAR ELLIPTIC SYSTEMS WITH DINI COEFFICIENTS UNDER NATURAL GROWTH CONDITION FOR THE CASE:1<m<2[J];Acta Mathematica Scientia;2012年05期

2 邱亞林;譚忠;;OPTIMAL INTERIOR PARTIAL REGULARITY FOR NONLINEAR ELLIPTIC SYSTEMS WITH DINI CONTINUOUS COEFFICIENTS[J];Acta Mathematica Scientia;2010年05期

3 ;The Method of A-Harmonic Approximation and Boundary Regularity for Nonlinear Elliptic Systems under the Natural Growth Condition[J];Acta Mathematica Sinica(English Series);2009年01期

4 鄭神州;;自然增長下的具VMO系數(shù)擬線性橢圓組的部分正則性[J];數(shù)學(xué)年刊A輯(中文版);2008年01期

5 袁秋寶;譚忠;;具有Dini連續(xù)性系數(shù)的非線性橢圓方程組在自然增長條件下的最優(yōu)內(nèi)部部分正則性(英文)[J];數(shù)學(xué)研究;2007年03期

6 陳淑紅;譚忠;;OPTIMAL INTERIOR PARTIAL REGULARITY FOR NONLINEAR ELLIPTIC SYSTEMS UNDER THE NATURAL GROWTH CONDITION:THE METHOD OF A-HARMONIC APPROXIMATION[J];Acta Mathematica Scientia;2007年03期

7 譚忠;王培林;李穎穎;;A-調(diào)和逼近方法和具有可控增長條件的非線性橢圓方程組最優(yōu)內(nèi)部部分正則性(英文)[J];數(shù)學(xué)研究;2007年01期

8 ;The method of p-harmonic approximation and optimal interior partial regularity for energy minimizing p-harmonic maps under the controllable growth condition[J];Science in China(Series A:Mathematics);2007年01期

，

本文編號(hào)：1498955