满足开集条件的自相似结构上的分析

In both theory and practice, to apply analysis methods to non-smooth geometric objects, we need to extend the traditional theory of analysis from manifolds to the metric measure spaces. Being one of the three common classes of metric measure spaces, fractal sets are considered as the best model for polymer, porous media and percolation in practice, and provide many instants and much inspiration for the analysis on general metric measure spaces in theory. The theory of analysis on fractals increases rapidly. Mathematicians provide the whole theory including the existence of Laplacian on fractal set, the function spaces on fractal set and heat kernel estimation. On the other hand, the theory of all the beginning, that is, the existence of Laplacian is limited in two narrow classes of fractal sets. In this project, we aim to extend the theory to a larger class, that is, self-similar structure satisfying open set condition, where the open set condition is the most significant separation condition in field of fractal geometry, which implies that the overlapping of different parts of a fractal set keeps small. Using the technique of Martin boundary, the self-similar structure, which is a generalization of self-similar set, can be related to a countable graph. Making use of fruitful results in spectral graph theory, we hope to define the Laplacian on the self-similar structure satisfying open set condition, establish the relation between the spectrum of Laplacian and geometric property of the graph, characterize the function spaces on self-similar structure, and estimate the heat kernel.

在理论和实践中，为了把分析方法应用于不光滑的几何对象，急需把传统流形上的分析理论推广到度量测度空间上。分形集作为现有的三种常见度量测度空间之一，不但在实践中是高分子物理、多孔介质和渗流的最好模型，而且在理论上也能为建立更一般的度量测度空间上的分析提供实例和启发。分形上的分析发展迅速，建立了从拉普拉斯算子的存在性、分形上的函数空间到热核估计的一整套理论，但是能够成功定义拉普拉斯算子的分形集却限制在两个比较窄的小类上。本项目试图把已有的理论推广到一个比较大的类，满足开集条件的自相似结构上，其中开集条件是分形几何领域最重要的分离条件，保证了分形集不同部分的重叠比较小。利用马丁边界的技巧，自相似结构（自相似集的推广）可以和一个可数无穷图相对应。借用图谱理论在图的分析中丰富的结果，我们希望定义自相似结构上的拉普拉斯算子、建立算子谱和图的几何性质之间的联系、刻画自相似结构上的函数空间、并估计热核。

分形上的分析，是把传统光滑空间(例如欧氏空间和流形)上的分析工具推广到复杂介质的数学模型(即分形)上。在这个方向上，亟待科研工作者解决的问题包括，拉普拉斯算子的存在性，分形上的函数空间理论，热核的估计。我们集中对“满足开集条件”的分形集展开以上问题的研究，给出了“开集条件”的刻画，并针对满足“倒向临界有限”条件的分形集，得到拉普拉斯算子的存在性以及热核的次高斯估计。