p.c.f.自相似集上的“热点”猜想及分形插值函数

发布时间：2018-04-19 08:21

  本文选题：p.c.f.自相似集 + “热点”猜想　； 参考：《浙江大学》2017年博士论文

【摘要】：本文主要包括两方面的工作.1. “热点”猜想.“热点”猜想是由J.Rauch于1974年提出的,其研究区域是欧式空间中的区域.2012年,利用谱提取算法,阮火军在Sierpinski垫片上证明了“热点”猜想成立.本文继续这方面的工作,在一些p.c.f.自相似集上考察“热点”猜想,证明了“热点”猜想在高维的Sierpinski垫片上成立,但是在六角垫片上不成立.从而得出,对于一般的p.c.f.自相似集“热点”猜想不成立.2.分形插值函数.首先对于定义在p.c.f自相似集上的分形插值函数,本文刻划了它们能量有限的充要条件.接着对于Sierpinski垫片(SG)上具有一致纵向尺度的分形插值函数,本文讨论了它们的Laplacian算子.作为应用,证明了以下Dirichlet问题的解为一致纵向尺度因子为1/5的分形插值函数:其中q1,q2,q3为SG的边界点,a1,a2,a3,η ∈R.最后对于SG上一类具有相同纵向尺度因子的分形插值函数,研究了它们的最值问题,得出它们与基本函数具有相同取值范围的充分必要条件.
[Abstract]:This paper mainly includes two aspects of work.Hot spot conjecture.The "hot spot" conjecture was put forward by J.Rauch in 1974 and its research area is in the European space. In 2012, using the spectral extraction algorithm, Ruan Huo Jun proved that the "hot spot" conjecture was true on the Sierpinski gasket.This article continues this work in some p.c.f.It is proved that the "hot spot" conjecture is true on the high-dimensional Sierpinski gasket, but not on the hexagonal gasket.Thus, for the general p.c.f.The "hot spot" conjecture of the self-similar set does not hold. 2.Fractal interpolation function.Firstly, for the fractal interpolation functions defined on the p.c.f self-similar set, the sufficient and necessary conditions for their finite energy are described in this paper.Then we discuss their Laplacian operators for fractal interpolation functions with uniform longitudinal scale on Sierpinski gasket.As an application, it is proved that the solution of the following Dirichlet problem is a fractal interpolation function with a uniform longitudinal scale factor of 1 / 5: where Q1Q2Q3 is the boundary point of SG, and 畏 鈭，

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