Differential Equations on Fractals

A Tutorial

By: Robert S. Strichartz

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Paperback | 31 October 2006

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192 Pages

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23.4 x 15.2 x 1.27

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Differential Equations on Fractals opens 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 lots of 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 Fractals explains 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.

Industry Reviews

"Well written and accessible to undergraduates or anybody who would like to obtain a quick but well-rounded introduction to fractal analysis. It is highly recommended and will certainly find a well-deserving place on many bookshelves."--Peter R. Massopust, Mathematical Reviews "The subject matter of this book is important to all mathematical scientists... Is this a good book for your library? It's better than that. Put this slim volume in your backpack next time you hiking by the sea."--Michael F. Barnsley, SIAM Review

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Introduction	p. vii
Measure, Energy, and Metric	p. 1
Graph Approximations	p. 1
Self-similar Measures	p. 5
Graph Energies	p. 10
Energy	p. 18
Electric Network Interpretation	p. 23
Effective Resistance Metric	p. 27
Notes and References	p. 29
Laplacian	p. 31
Weak Formulation	p. 31
Pointwise Formula	p. 34
Normal Derivatives	p. 37
Gauss-Green Formula	p. 41
Gluing	p. 44
Green's Function	p. 46
Local Behavior of Functions	p. 55
Notes and References	p. 62
Spectrum of the Laplacian	p. 63
Fourier Series Revisited	p. 63
Spectral Decimation	p. 68
Eigenvalues and Multiplicities	p. 73
Localized Eigenfunctions	p. 79
Spectral Asymptotics	p. 83
Integrals Involving Eigenfunctions	p. 86
Notes and References	p. 89
Postcritically Finite Fractals	p. 91
Definitions	p. 91
Energy Restriction and Renormalization	p. 96
Examples	p. 101
Laplacians	p. 109
Geography Is Destiny	p. 114
Non-self-similar Fractals	p. 116
Notes and References	p. 119
Further Topics	p. 121
Polynomials, Splines, and Power Series	p. 121
Local Symmetries	p. 125
Energy Measures	p. 127
Fractal Blow-ups and Fractafolds	p. 131
Singularities	p. 136
Products of Fractals	p. 140
Solvability of Differential Equations	p. 146
Heat Kernel Estimates	p. 149
Convergence of Fourier Series	p. 152
Notes and References 156 References 159 Index	p. 167
Table of Contents provided by Publisher. All Rights Reserved.

Differential Equations on Fractals

A Tutorial

At a Glance

Paperback

Industry Reviews

Shipping

How to return your order

Defective items

You Can Find This Book In

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