AI 中文总结
本文用超图拉普拉斯算子替换Cahn-Hilliard方程中的拉普拉斯算子，提出超图上的扩散模型，并利用超图拉普拉斯性质给出新证明方法。
AI 中文摘要
为研究以超图形式表示的网络结构，引入了超图拉普拉斯算子。结果表明，与超图拉普拉斯算子相关的演化方程的解的行为与经典热方程的解非常相似。因此，通过将偏微分方程中的拉普拉斯算子替换为超图拉普拉斯算子，我们或许能够在超图（离散域）上引入各种扩散模型，其解的行为与偏微分方程相似。在本文中，我们考虑通过将原始Cahn-Hilliard方程中的拉普拉斯算子替换为超图拉普拉斯算子而得到的方程组。由于超图拉普拉斯算子的非线性和多值性，很难将原始Cahn-Hilliard方程的方法应用于我们的问题。为了应对这些困难，本文将利用超图拉普拉斯算子的性质引入一种新的证明方法。
英文摘要
A hypergraph Laplacian was introduced to investigate the structure of networks written as hypergraphs. It was shown that the behavior of solutions to an evolution equation associated with the hypergraph Laplacian quite resembles that of solutions to the classical heat equation. Hence by replacing the Laplacian in PDEs with the hypergraph Laplacian, we might be able to introduce various diffusion models on hypergraphs (discrete domains), whose behavior of solution is similar to PDEs. In this paper, we consider a system of equations obtained by replacing the Laplacian in the original Cahn--Hilliard equation with the hypergraph Laplacian. Due to the nonlinearity and multivaluedness of the hypergraph Laplacian, it is difficult to apply methods for the original Cahn--Hilliard equation to our problem. To cope with these difficulty, we shall introduce a new proof by using properties of the hypergraph Laplacian in this paper.
Comments40 pages, 2 figures
