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基本説明
Focuses on the construction of a Laplacian on the Sierpinsky gasket and related fractals.
Full Description
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.
Contents
Introduction vii Chapter 1. Measure, Energy, and Metric 1 1.1 Graph Approximations 1 1.2 Self-similar Measures 5 1.3 Graph Energies 10 1.4 Energy 18 1.5 Electric Network Interpretation 23 1.6 Effective Resistance Metric 27 1.7 Notes and References 29 Chapter 2. Laplacian 31 2.1 Weak Formulation 31 2.2 Pointwise Formula 34 2.3 Normal Derivatives 37 2.4 Gauss-Green Formula 41 2.5 Gluing 44 2.6 Green's Function 46 2.7 Local Behavior of Functions 55 2.8 Notes and References 62 Chapter 3. Spectrum of the Laplacian 63 3.1 Fourier Series Revisited 63 3.2 Spectral Decimation 68 3.3 Eigenvalues and Multiplicities 73 3.4 Localized Eigenfunctions 79 3.5 Spectral Asymptotics 83 3.6 Integrals Involving Eigenfunctions 86 3.7 Notes and References 89 Chapter 4. Postcritically Finite Fractals 91 4.1 Definitions 91 4.2 Energy Restriction and Renormalization 96 4.3 Examples 101 4.4 Laplacians 109 4.5 Geography Is Destiny 114 4.6 Non-self-similar Fractals 116 4.7 Notes and References 119 Chapter 5. Further Topics 121 5.1 Polynomials, Splines, and Power Series 121 5.2 Local Symmetries 125 5.3 Energy Measures 127 5.4 Fractal Blow-ups and Fractafolds 131 5.5 Singularities 136 5.6 Products of Fractals 140 5.7 Solvability of Differential Equations 146 5.8 Heat Kernel Estimates 149 5.9 Convergence of Fourier Series 152 5.10 Notes and References 156 References 159 Index 167
