Dirac调和系统及其在算子理论中的应用

本项目以带有Hodge-Dirac算子的调和系统为研究对象，拟对该系统解的分析性质进行深入研究。这类调和系统是一类微分流形上的偏微分方程，其解空间是基于微分形式定义的。微分形式是欧式空间中可微函数的衍生与推广，它在几何分析和偏微分方程理论的发展中起着十分重要的作用，是研究流形上各类偏微分方程的有力工具。近些年，非线性弹性理论、理论物理学的发展极大的推动了微分形式的偏微分系统研究。目前，微分形式椭圆型方程已成为了该领域研究的热点问题之一。.本项目旨对该系统的解进行定性和定量分析，主要包括：建立基于不同解空间的Caccioppoli不等式，Sobolev嵌入不等式等；尝试对方程弱解和很弱解的适定性、hölder连续性等进行系统地研究；实现基于该方程解在算子有界性证明中的应用。本项目的创新之处在于研究对方程非齐次项算子进行了分情况讨论，揭示该方程的数学意义及在算子理论中发挥的理论作用和优势。

This project aims at a harmonic equation driven by the Hodge-Dirac operator and intends to explore the analysis properties of solutions to this system deeply. This system involves some kinds of partial differential equations on differential manifolds, in which the solutions are defined on the differential form space. Differential form is the variant and generalization of differential functions on Euclidean space, and it plays a very important role in the research for geometric analysis and partial differential equations since it is a quite powerful tool for the study on some partial differential systems via differential manifolds. In recent years, due to the development of nonlinear elastic theory and theoretical physics, partial differential system for differential forms have achieved significant progress. Now, the investigation on elliptic equations for differential forms is a hot topic in harmonic analysis and its related fields..This project focuses on the quantitative and qualitative analysis of solutions. In precise, it will establish Caccioppoli inequalities, Sobolev-imbedding inequalities on some given spaces of differential forms. Then, by making use of the proven basic inequalities, it proceed to the study on the properties of the weak and very weak solutions to Dirac-harmonic system, including the wellposedness,Hölder continuity. Moreover, we will illustrate the boundedness of the operators by applying the proven results. The innovation of this project is that it discusses the nonlinear terms by different cases and reveals the mathematical relations between operator theory and Dirac-harmonic equation for differential forms.