p-Laplacian特征值的基本间隙研究

本项目研究p-Laplacian 的基本间隙（前两个特征值之差），聚焦于边界光滑且具有紧闭包的区域上基本间隙的上下界估计。研究内容包括：（1）Dirichlet p-Laplacian基本间隙的最优下界；（2）Neumann p-Laplacian基本间隙的最优上界；（3）离散空间(树)上的Neumann Laplacian的最优上界。核心思想是：利用连续模方法及移植平均法，建立所研究的区域与模型上 p-Laplacian的基本间隙之间的比较关系，得到该区域上p-Laplacian的基本间隙最优上下界的一个等价刻画。

In this project, we consider the fundamental gap (the difference between the first two eigenvalues) of the p-Laplacian, mainly focus on the upper(optimal)/lower bounds estimates of the p-Laplacian on compact domains with smooth boundary. The main contents includes:(1) the lower bound of the fundamental gap of Dirichlet p-Laplacian; (2)the optimal upper bound of the fundamental gap of Neumann p-Laplacian; (3)the optimal upper bound of Neumann Laplacian on discrete spaces(trees). The key ideas are as follows: To derive an equivalent characterization of the optimal upper and lower bounds of the fundamental gap for the p-Laplacian, by establishing a fundamental gap comparison relationship between the compact domains and that of their corresponding models. The main methods involve the method of modulus of continuity and the transplanting and averaging method.