基于霍奇理论的外微分形式拉普拉斯迭代方程解的研究

Study on Solutions of External Differential Laplace Iterative Equations Based on Hodge’s Theory

摘要: 本文主要研究黎曼流形上外微分形式拉普拉斯方程解的存在性问题。首先利用Hodge分解定理和格林算子对Laplace算子Δ的无穷迭代特征值谱分析给出求解方法，然后利用迭代法对外微分形式的拉普拉斯方程Δα=ω进行拓展，最后得到k阶拉普拉斯方程的迭代解，这对高数阶的拉普拉斯方程的解和特征值谱分析的研究产生了一定的影响。

Abstract: In this paper, we mainly study the existence of solutions to the external differential form Laplace’s equation on Riemannian manifold. Firstly, the Hodge decomposition theorem and the Green oper-ator are used to give a solution method for the infinite iterative eigenvalue spectrum analysis of the Laplace operator. Then, the iterative method is used to expand the Laplace’s equation Δα=ω in the external differential form, and finally the iterative solution of the Laplace’s equation k-order is obtained, which has a certain impact on the research of the solutions of the Laplace’s equation of higher order and eigenvalue spectrum analysis.