p.c.f.自相似集上的热点猜想及分形插值函数

The main research is focused on the hot spots conjecture and fractal interpolation functions on p.c.f. self-similar sets..1. Hot spots conjecture. The hot spots conjecture was posed by J.Rauch in 1974, while, until recently, the conjecture has been proved only for a very limited family of domains. This project studies the hot spots conjecture on the following two types of p.c.f. self-similar sets: hexagasket, whose boundary include all the non-junction points and some fractal domains with infinite boundary. Furthermore we consider what qualities of p.c.f. self-similar sets make the conjecture true ..2. Fractal interpolation functions (FIFs). In the previous study, we have provided the necessary and sufficient condition such that FIFs on p.c.f. self-similar sets have finite energy. In this project, we research on the conditions where the Laplacian of FIFs is finite. To solve the problem, we start with FIFs on Sierpinski gasket and FIFs on p.c.f. self-similar sets with the same vertical scaling factors, and then generalize the research scope gradually..The project contributes to a better comprehension of Laplacian on fractals and provides a testing ground for analysis of fractals.

本项目主要研究p.c.f.自相似集上的热点猜想以及分形插值函数。.1.热点猜想。热点猜想是由J.Rauch于1974年提出的，然而直到今天，此猜想仅仅在非常有限的区域上得到了证明。本项目主要在以下两类p.c.f.自相似集上考察热点猜想：将所有的非连接点都定义为边界点的六角垫片以及具有无限边界点的分形集。从而进一步探究具备什么条件的p.c.f.自相似集，热点猜想成立。.2.分形插值函数。前期的研究已经给出了p.c.f.自相似集上分形插值函数能量有限的充分必要条件，本项目进一步研究其在什么条件下具有有限的Laplacian。我们从Sierpinski垫片上的分形插值函数以及一般的p.c.f.自相似集上具有相同纵向尺度因子的分形插值函数入手，逐步将研究对象一般化，从而解决问题。.本项目有助于更加深刻地理解分形上的Laplacian算子，并为分形上的分析提供更多的研究函数。

该项目验证了热点猜想在vicsek 集上成立，并讨论了分形插值函数的反Holder条件，以及高维的递归分形插值函数关于第一个变量压缩条件的限制。