Differential Equations on Fractals

by Strichartz, Robert S.

• ISBN13: 9780691127316
• ISBN10: 069112731X
• Format: Paperback
• Copyright: 2006-07-31
• Publisher: Princeton Univ Pr

Summary

Differential Equations on Fractalsopens the door to understanding the recently developed area of analysis on fractals, focusing on the construction of a Laplacian on the Sierpinski gasket and related fractals. Written in a lively and informal style, with numerous intriguing exercises on all levels of difficulty, the book is accessible to advanced undergraduates, graduate students, and mathematicians who seek an understanding of analysis on fractals. Robert Strichartz takes the reader to the frontiers of research, starting with carefully motivated examples and constructions. One of the great accomplishments of geometric analysis in the nineteenth and twentieth centuries was the development of the theory of Laplacians on smooth manifolds. But what happens when the underlying space is rough? Fractals provide models of rough spaces that nevertheless have a strong structure, specifically self-similarity. Exploiting this structure, researchers in probability theory in the 1980s were able to prove the existence of Brownian motion, and therefore of a Laplacian, on certain fractals. An explicit analytic construction was provided in 1989 by Jun Kigami.Differential Equations on Fractalsexplains Kigami's construction, shows why it is natural and important, and unfolds many of the interesting consequences that have recently been discovered. This book can be used as a self-study guide for students interested in fractal analysis, or as a textbook for a special topics course.

Table of Contents

Introduction	vii
Measure, Energy, and Metric	1	(30)
Graph Approximations	1	(4)
Self-similar Measures	5	(5)
Graph Energies	10	(8)
Energy	18	(5)
Electric Network Interpretation	23	(4)
Effective Resistance Metric	27	(2)
Notes and References	29	(2)
Laplacian	31	(32)
Weak Formulation	31	(3)
Pointwise Formula	34	(3)
Normal Derivatives	37	(4)
Gauss--Green Formula	41	(3)
Gluing	44	(2)
Green's Function	46	(9)
Local Behavior of Functions	55	(7)
Notes and References	62	(1)
Spectrum of the Laplacian	63	(28)
Fourier Series Revisited	63	(5)
Spectral Decimation	68	(4)
Eigenvalues and Multiplicities	72	(7)
Localized Eigenfunctions	79	(4)
Spectral Asymptoties	83	(3)
Integrals Involving Eigenfunctions	86	(3)
Notes and References	89	(2)
Postcritically Finite Fractals	91	(30)
Definitions	91	(5)
Energy Restriction and Renormalization	96	(5)
Examples	101	(8)
Laplacians	109	(5)
Geography Is Destiny	114	(2)
Non-self-similar Fractals	116	(3)
Notes and References	119	(2)
Further Topics	121	(38)
Polynomials, Splines, and Power Series	121	(4)
Local Symmetries	125	(2)
Energy Measures	127	(4)
Fractal Blow-ups and Fractafolds	131	(5)
Singularities	136	(4)
Products of Fractals	140	(6)
Solvability of Differential Equations	146	(3)
Heat Kernel Estimates	149	(3)
Convergence of Fourier Series	152	(4)
Notes and References	156	(3)
References	159	(8)
Index	167