This paper is concerned with the derivation and properties of differential complexes arising from a variety of problems in differential equations, with applications in continuum mechanics, relativity, and other fields. We present a systematic procedure which, starting from well-understood differential complexes such as the de Rham complex, derives new complexes and deduces the properties of the new complexes from the old. We relate the cohomology of the output complex to that of the input complexes and show that the new complex has closed ranges, and, consequently, satisfies a Hodge decomposition, Poincaré-type inequalities, well-posed Hodge–Laplacian boundary value problems, regular decomposition, and compactness properties on general Lipschitz domains.

本文关注由微分方程中的各种问题引起的微分络合物的推导和性质，及其在连续力学，相对论和其他领域中的应用。我们提出了一种系统的程序，该程序从易于理解的微分配合物（例如de Rham配合物）开始，衍生出新配合物并从旧推论出新配合物的性质。我们将输出复合物的同调与输入复合物的同调相关联，并表明新复合物具有封闭范围，因此，满足Hodge分解，Poincaré型不等式，摆放良好的Hodge-Laplacian边值问题，正则分解，以及一般Lipschitz域上的紧实度属性。