Dirac-调和方程解的性质及其相关算子的范数估计

发布时间：2018-02-03 07:59

  本文关键词： 微分形式 Dirac-调和方程 Orlicz-Sobolev嵌入不等式 弱逆H?lder不等式 算子 范数估计　出处：《哈尔滨工业大学》2017年博士论文　论文类型：学位论文

【摘要】：近些年,非线性弹性理论和拟共形映射的发展促使微分形式椭圆方程的研究取得了极大的进展,已经从最初的Laplace方程扩展到了 A-调和方程。Hodge-Dirac算子的发展来源于理论物理学,它不仅在量子力学和广义相对论中有着不可替代的作用,而且为几何学和代数学的研究提供了有力的数学工具。2015年,Ding和Liu基于Hodge-Dirac算子和齐次A-调和方程,提出了齐次Dirac-调和方程及其弱解的概念,促进了 A-调和方程的进一步发展,也使得Hodge-Dirac算子有了更广泛的应用。作为A-调和方程的衍生方程,齐次Dirac-调和方程的理论研究仍处于起步阶段,其数学意义和实际作用还有待深入了解。因此,本文重点讨论了微分形式的Dirac-调和方程解的性质及其在相关算子中的理论应用,其主要研究内容为:首先,为了研究齐次Dirac-调和方程在复合算子理论中的实际作用,本文讨论了基于齐次Dirac-调和方程解的两类复合算子的有界性问题。由于Poincare不等式和Orlicz-Sobolev嵌入不等式在建立算子有界性理论中有着根本性的作用,所以本文主要利用齐次Dirac-调和方程解的基本不等式Lφ-平均域的性质,通过选取一类特殊的Young函数,证明了齐次Dirac-调和方程解的复合算子的Poincare型不等式和Orlicz-Sobolev嵌入不等式,由此得到了复合算子依Orlicz范数和Orlicz-Sobolev范数的有界性。其次,本文受Possion方程中复合算子D2G的启发引入了两类新的迭代算子D~kG~k和D~(k+1)G~k,并对该类迭代算子的高阶可积性及其依BMO范数和局部Lipschitz范数的有界性问题进行了研究。虽然Green算子及其梯度经多次复合后仍具有很好的可积性,但是由于Hodge-Dirac算子与外微分算子有关,这使得建立迭代算子高阶可积性的难度增大。为克服这一难点,本文将微分形式的Poincare-Sobolev不等式作为关键工具,通过构造与指数p和空间维数nn有关的辅助参数,建立了 1pn和p≥n两种情况下的迭代算子的高阶可积性。在此基础上,本文借助L~p空间的Hodge分解定理,得到了迭代次数kk取奇数和偶数时两类迭代算子最简化的表达式,该表达式为建立范数比较定理起到了决定性作用。最后,在一定基础性假设条件下,本文提出了非齐次Dirac-调和方程及其弱解的概念,并研究了该方程解的基本性质。非齐次Dirac-调和方程与齐次Dirac-调和方程的区别在于非齐次项部分,而正是这一部分使得研究变的复杂。为解决这一难点,本文根据研究的需要,对非齐次Driac-调和方程中算子A和B适当的附加了一些结构性约束条件,进而利用微分形式的Lp理论建立了非齐次Dirac-调和方程解的收敛性和解的基本不等式,包括:弱逆Holder不等式,Caccioppoli不等式和Orlicz-Sobolev嵌入不等式。与此同时,文本借助Hodge分解定理和一定的处理技巧,构造了非线性有界算子,并利用Minty-Browder定理得到了 一类具体的非齐次Dirac-调和方程解的存在唯一性定理。
[Abstract]:In recent years, the development of nonlinear elastic theory and quasi-conformal mapping has made great progress in the study of differential elliptic equations. The development of the Hodge-Dirac operator from the initial Laplace equation to the A- harmonic equation .Hodge-Dirac operator is derived from theoretical physics. It not only plays an irreplaceable role in quantum mechanics and general relativity, but also provides a powerful mathematical tool for the study of geometry and algebra. 2015. Ding and Liu put forward the concept of homogeneous Dirac-harmonic equation and its weak solution based on Hodge-Dirac operator and homogeneous A-harmonic equation. It promotes the further development of the A- harmonic equation and makes the Hodge-Dirac operator more widely used as the derivative equation of the A- harmonic equation. The theoretical study of homogeneous Dirac-harmonic equations is still in its infancy, and its mathematical significance and practical function need to be deeply understood. In this paper, we mainly discuss the properties of the solutions of Dirac-harmonic equations in differential form and their theoretical applications in related operators. The main research contents are as follows: first of all. In order to study the practical function of homogeneous Dirac-harmonic equation in the theory of composition operator. In this paper, we discuss the boundedness of two kinds of composition operators based on the solutions of homogeneous Dirac-harmonic equations. The Poincare inequality and Orlicz-Sobolev embedding inequality are established. The theory of operator boundedness plays a fundamental role. In this paper, we mainly use the properties of L 蠁 -mean domain of the fundamental inequality of solutions of homogeneous Dirac-harmonic equations by selecting a special class of Young functions. The Poincare type inequality and Orlicz-Sobolev embedding inequality for composition operator of homogeneous Dirac-harmonic equation are proved. The boundedness of composition operators according to Orlicz norm and Orlicz-Sobolev norm is obtained. Secondly. In this paper, inspired by the composition operator D2G in the Possion equation, two kinds of new iterative operators DKG ~ (K) and D ~ (+) K ~ (1) G ~ (1) are introduced. The higher order integrability of this kind of iterative operators and the boundedness of Green operators based on BMO norm and local Lipschitz norm are studied. Although the Green operator and its gradient are composed several times, they still have. Good integrability. But because the Hodge-Dirac operator is related to the exterior differential operator, it is more difficult to establish the higher order integrability of the iterative operator. In this paper, Poincare-Sobolev inequality in differential form is used as a key tool to construct auxiliary parameters related to exponent p and space dimension n n. The higher order integrability of iterative operators in the case of 1pn and p 鈮，

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