黎曼流形上的微分形式(英文)

Differential Forms on Riemannian Manifolds

黎曼流形上弱闭微分形式的WT-类是由D.Franke等引入并研究的.它们密切联系于椭圆型偏微分方程和拟正则映射的正则性理论,并在空间几何函数论中充当重要角色.为了研究黎曼流形上的弱闭微分形式,我们首先引进WT2-类弱闭微分形式的定义,然后通过选取适当的测试函数θ并仿照D.Franke等的证明方法,证明了若w为非齐次调和方程δA( m,dw)=δh( m,dw)的解,这里A和h满足增长条件和强制性条件,则dw属于WT2-类.这一结果可认为是D.Franke等结果的推广.它说明了微分形式的WT类与非齐次微分方程有密切关系.由主要定理和WT2-类微分形式的性质可推出Caccioppoli不等式,再结合Gehring引理就有正则性结果.

WT- classes of weakly closed differential forms on Riemannian manifolds are introduced and studied by D. Franke, et al. They are closely related to the regularity theory of the elliptic partial differential equations and quasi-regular mappings, and serve as an important role in the study of geometric function theory in space. The aim of the present paper is to study the weakly closed differential forms on Riemannian manifolds. We first introduce the definition of WT2 class of differential forms, and then by imitating the method of D. Franke et al. , and choosing an appropriate test function θ, prove that if n is a solution of the non - homogeneous harmonic equation δA （m,dw） =δh （m,dw）, where A and h satisfy some growth and coercive conditions, then dw is in the class WT2. This result can be regarded as a generalization of the known result, which illustrates the fact that the WT - classes of the differential forms are closely connected with non-homogeneous differential equations. From the main theorem and the properties of WT2 differential forms one can deduce a Caccioppoli inequality, which is associated with Gehring＇s Lemma yields a regularity result.