某些分形集上拉普拉斯算子的谱分析及相关问题

As a theoretical corner stone of fractal differential equations, the spectral theory of fractal Laplacians is one of the most hot branch of current fractal analysis. In essence the analysis on spectrum of Laplacian reflects certain intrinsic analytical qualities and geometric characteristics of fractal objects. It has great theoretical merit for the development of fractal analysis and especially the theory of fractal differential equations. This project, firstly, studies the exact spectrum and the Weyl asymptotic property of the fractal Laplacian on p.c.f. self-similar fractal domains and some sub-domains with true fractal boundaries. The first and second term estimates of the eigenvalue counting function will be analyzed. The relationship between Weyl's asymptotic property and the dimensions of fractal domains and their boundaries will be found. The Weyl-Berry's conjecture on the Laplacian will be extended to the fractal case. Secondly, we will extend the results to some non p.c.f. self-similar sets which have strong symmetric properties, characterize the general properties of the fractal spectrum and reveal the essential differences of the spectral theory between fractal case and classical case. Moreover, the deep connection to all the other relevant topics in fractal analysis theory will also be studied. Finally, since the local fields have natural hierarchical and fractal-like structures, we will develop the pseudo-differential operator theory on local fields, especially the spectrum theory of the Lapalcian. The connection of the local fields analysis to theory of fractal analysis will be pursued in order to establish the foundation for fractal dynamics.

作为分形微分方程的核心理论支撑，分形Laplace算子的谱理论是当前分形分析领域中最热门的研究分支之一。Laplace算子的谱从本质上反映了所研究分形对象的固有分析属性与几何特征，对发展分形分析，特别是分形微分方程理论具有重要理论价值。本项目首先研究p.c.f.自相似分形区域及某些具有真正分形边界的子区域上分形Laplace算子的精确谱刻划及谱的Weyl渐近性质，分析特征值计数函数的主项与二阶项估计，寻求谱渐近律与相应分形区域及边界维数的联系，验证Weyl-Berry 猜测的分形样式。然后推广到某些具有强对称性的非p.c.f.自相似集合上，刻划谱的一般性质，揭示分形Laplace算子谱理论与经典情形的本质异同，探求谱理论与分形分析理论相关课题的深刻联系。最后利用局部域天然的分级结构，发展局部域上拟微分算子，特别是Laplace 算子，寻求它与分形分析理论的联系，为建立分形动力学打下基础。

作为分形微分方程的核心理论支撑，分形Laplace算子的谱理论是当前分形分析领域中最热门的研究分支之一。Laplace算子的谱从本质上反映了所研究分形对象的固有分析属性与几何特征，对发展分形分析，特别是分形微分方程理论具有重要理论价值。本项目首先研究p.c.f.自相似分形区域及某些具有真正分形边界的子区域上分形 Laplace 算子的精确谱刻划及谱的Weyl 渐近性质，分析特征值计数函数的主项与二阶项估计，寻求谱渐近律与相应分形区域及边界维数的联系，验证Weyl-Berry猜测的分形样式。然后推广到某些具有强对称性的非p.c.f.自相似集合上，刻划谱的一般性质，揭示分形Laplace算子谱理论与经典情形的本质异同，探求谱理论与分形分析理论相关课题的深刻联系。最后利用局部域天然的分级结构，发展局部域上拟微分算子，特别是Laplace 算子，寻求它与分形分析理论的联系，为建立分形动力学打下基础。