Some Properties Of Operators Based On Differential Forms And Integrability Of Solutions To Harmonic Equations

Differential forms were firstly used to establish calculus theory on manifolds, which unifies Newton-Leibiniz formula,Green formula,Gauss formula and Stokes formula of the classical calcu-lus to Stokes formula on manifolds. Thus differential form is a basic tool in modern analysis, and has a wide range of applications in many areas such as modern differential geometry, differential equa-tion, group theory and physics. In this dissertation, we deal with some properties of the Caratheodory operator of differential forms, the prior estimates of a generatized A-harmonic equation, the Poincare Lemma for a more generalized domain, and the corresponding homotopy operator related to this do-main. For generalized A-harmonic equations for differential forms, we gain the prior estimates for their solutions. Under a more relaxed condition than convex domain, we gain an extension of the Poincare Lemma of differential forms, and the homotopy operator can be used in a kind of more universal domains. This dissertation consists of four chapters and the main contents are as follows:In the first chapter, we preasent a survey to the development of A-harmonic equations of differ-ential forms and the background of differential forms. Meanwhile, the main results in this dissertation are summarized.In chapter two, as A-harmonic tensor is a special kind of Caratheodory operator, we extend the Caratheodory operator to the differential forms. We find that A-harmonic tensors have some structure very similar to the Caratheodory operator of functions. With the study of Caratheodory operator for functions, we establish Caratheodory operator for differential forms and prove it is still bounded and continuous. Then we characterize A-harmonic tensors by the Caratheodory operator based on differential forms.In the chapter three we obtain important inequalities for a kind of generalized A-harmonic equa-tions, including the reverse Holder inequalities for the solution and its differential and Caccioppoli inequality. In the prior estimates, as we know, Holder inequality has important position and is often used to get some better estimates. Thus we study the Holder inequalities for these equations, which can be reduced to the existing ones.In chapter four we obtain an extension of the Poincare Lemma of differential forms and some estimates of the composition of the homotopy and Caratheodory operators. Poincare Lemma is an important result, which helps us learn better about the differential forms. And homotopy operator is also very important for differential forms. What is most important in our result is we use the contraction by smooth curve instead of by straight-line path, which extend the applied domain.