HARMONIC FUNCTIONS AND THE SPECTRUM OF THE LAPLACIAN ON THE SIERPINSKI CARPET

被引：11
作者：
Begue, Matthew [1 ]
Kalloniatis, Tristan [2 ]
Strichartz, Robert S. [3 ]
机构：
[1] Univ Maryland, Dept Math, College Pk, MD 20742 USA
[2] Univ Cambridge, Fac Math, Cambridge CB3 0WA, England
[3] Cornell Univ, Dept Math, Ithaca, NY 14853 USA
来源：
FRACTALS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY | 2013年 / 21卷 / 01期
基金：
美国国家科学基金会;
关键词：
Sierpinski Carpet; Fractal; Laplacian; Spectrum; Poisson Kernel; Heat Kernel; Dirichlet Kernel; Covering Spaces; Fractafolds; Normal Derivatives; RESISTANCE; FRACTALS;
D O I：
10.1142/S0218348X13500023
中图分类号：
O1 [数学];
学科分类号：
0701 ; 070101 ;
摘要：
Kusuoka and Zhou have defined the Laplacian on the Sierpinski carpet using average values of functions on small cells and the graph structure of cell adjacency. We have implemented an algorithm that uses their method to approximate solutions to boundary value problems. As a result we have a wealth of data concerning harmonic functions with prescribed boundary values, and eigenfunctions of the Laplacian with either Neumann or Dirichlet boundary conditions. We will present some of this data and discuss some ideas for defining normal derivatives on the boundary of the carpet.
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页数：32
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