本文選題：p.c.f.自相似集 + “熱點(diǎn)”猜想��； 參考：《浙江大學(xué)》2017年博士論文

【摘要】：本文主要包括兩方面的工作.1. “熱點(diǎn)”猜想.“熱點(diǎn)”猜想是由J.Rauch于1974年提出的,其研究區(qū)域是歐式空間中的區(qū)域.2012年,利用譜提取算法,阮火軍在Sierpinski墊片上證明了“熱點(diǎn)”猜想成立.本文繼續(xù)這方面的工作,在一些p.c.f.自相似集上考察“熱點(diǎn)”猜想,證明了“熱點(diǎn)”猜想在高維的Sierpinski墊片上成立,但是在六角墊片上不成立.從而得出,對(duì)于一般的p.c.f.自相似集“熱點(diǎn)”猜想不成立.2.分形插值函數(shù).首先對(duì)于定義在p.c.f自相似集上的分形插值函數(shù),本文刻劃了它們能量有限的充要條件.接著對(duì)于Sierpinski墊片(SG)上具有一致縱向尺度的分形插值函數(shù),本文討論了它們的Laplacian算子.作為應(yīng)用,證明了以下Dirichlet問題的解為一致縱向尺度因子為1/5的分形插值函數(shù):其中q1,q2,q3為SG的邊界點(diǎn),a1,a2,a3,η ∈R.最后對(duì)于SG上一類具有相同縱向尺度因子的分形插值函數(shù),研究了它們的最值問題,得出它們與基本函數(shù)具有相同取值范圍的充分必要條件.
[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，

本文編號(hào)：1772320