本文關(guān)鍵詞： 微分形式 Dirac-調(diào)和方程 Orlicz-Sobolev嵌入不等式 弱逆H?lder不等式 算子 范數(shù)估計(jì)　出處：《哈爾濱工業(yè)大學(xué)》2017年博士論文　論文類型：學(xué)位論文

【摘要】：近些年,非線性彈性理論和擬共形映射的發(fā)展促使微分形式橢圓方程的研究取得了極大的進(jìn)展,已經(jīng)從最初的Laplace方程擴(kuò)展到了 A-調(diào)和方程。Hodge-Dirac算子的發(fā)展來源于理論物理學(xué),它不僅在量子力學(xué)和廣義相對(duì)論中有著不可替代的作用,而且為幾何學(xué)和代數(shù)學(xué)的研究提供了有力的數(shù)學(xué)工具。2015年,Ding和Liu基于Hodge-Dirac算子和齊次A-調(diào)和方程,提出了齊次Dirac-調(diào)和方程及其弱解的概念,促進(jìn)了 A-調(diào)和方程的進(jìn)一步發(fā)展,也使得Hodge-Dirac算子有了更廣泛的應(yīng)用。作為A-調(diào)和方程的衍生方程,齊次Dirac-調(diào)和方程的理論研究仍處于起步階段,其數(shù)學(xué)意義和實(shí)際作用還有待深入了解。因此,本文重點(diǎn)討論了微分形式的Dirac-調(diào)和方程解的性質(zhì)及其在相關(guān)算子中的理論應(yīng)用,其主要研究?jī)?nèi)容為:首先,為了研究齊次Dirac-調(diào)和方程在復(fù)合算子理論中的實(shí)際作用,本文討論了基于齊次Dirac-調(diào)和方程解的兩類復(fù)合算子的有界性問題。由于Poincare不等式和Orlicz-Sobolev嵌入不等式在建立算子有界性理論中有著根本性的作用,所以本文主要利用齊次Dirac-調(diào)和方程解的基本不等式Lφ-平均域的性質(zhì),通過選取一類特殊的Young函數(shù),證明了齊次Dirac-調(diào)和方程解的復(fù)合算子的Poincare型不等式和Orlicz-Sobolev嵌入不等式,由此得到了復(fù)合算子依Orlicz范數(shù)和Orlicz-Sobolev范數(shù)的有界性。其次,本文受Possion方程中復(fù)合算子D2G的啟發(fā)引入了兩類新的迭代算子D~kG~k和D~(k+1)G~k,并對(duì)該類迭代算子的高階可積性及其依BMO范數(shù)和局部Lipschitz范數(shù)的有界性問題進(jìn)行了研究。雖然Green算子及其梯度經(jīng)多次復(fù)合后仍具有很好的可積性,但是由于Hodge-Dirac算子與外微分算子有關(guān),這使得建立迭代算子高階可積性的難度增大。為克服這一難點(diǎn),本文將微分形式的Poincare-Sobolev不等式作為關(guān)鍵工具,通過構(gòu)造與指數(shù)p和空間維數(shù)nn有關(guān)的輔助參數(shù),建立了 1pn和p≥n兩種情況下的迭代算子的高階可積性。在此基礎(chǔ)上,本文借助L~p空間的Hodge分解定理,得到了迭代次數(shù)kk取奇數(shù)和偶數(shù)時(shí)兩類迭代算子最簡(jiǎn)化的表達(dá)式,該表達(dá)式為建立范數(shù)比較定理起到了決定性作用。最后,在一定基礎(chǔ)性假設(shè)條件下,本文提出了非齊次Dirac-調(diào)和方程及其弱解的概念,并研究了該方程解的基本性質(zhì)。非齊次Dirac-調(diào)和方程與齊次Dirac-調(diào)和方程的區(qū)別在于非齊次項(xiàng)部分,而正是這一部分使得研究變的復(fù)雜。為解決這一難點(diǎn),本文根據(jù)研究的需要,對(duì)非齊次Driac-調(diào)和方程中算子A和B適當(dāng)?shù)母郊恿艘恍┙Y(jié)構(gòu)性約束條件,進(jìn)而利用微分形式的Lp理論建立了非齊次Dirac-調(diào)和方程解的收斂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。與此同時(shí),文本借助Hodge分解定理和一定的處理技巧,構(gòu)造了非線性有界算子,并利用Minty-Browder定理得到了 一類具體的非齊次Dirac-調(diào)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，

本文編號(hào)：1486850