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Differential equations on fractals. A tutorial. (English) Zbl 1190.35001
Princeton, NJ: Princeton University Press (ISBN 0-691-12731-X/pbk; 0-691-12542-2/hbk). xiv, 169 p. (2006).
This tutorial provides a very simple introduction into the field of Analysis on fractals. The author – one of the most established and respected experts in this field – explains how to build up the notion of a Laplacian on finitely ramified fractals in a quite intuitive way. The presented approach deeply relies on the self-similarity and the finite ramification of the sets under consideration. The main idea is explained with the help of two examples, namely the unit interval \([0,1]\) and the Sierpinski gasket. The unit interval? Most people would not regard this set as a fractal. However, it is a finitely ramified self-similar set and can be approximated by an increasing sequence of countable sets of points (the dyadic numbers) which serve as vertex sets of a corresponding sequence of nested graphs on which some kind of “pre-Laplacian” is defined. More precisely, the construction is done by means of Dirichlet forms and via harmonic extensions of functions. It is shown that this approach leads to the classical notions of calculus on the interval. Hence, the presented approach extends the established analysis for one-dimensional domains to more general sets of possibly higher Hausdorff dimension.
The book consists of five chapters. We will now give an a bit more detailed description of them.
In the first chapter, basic notation and notions are provided, in particular self-similar sets and measures are introduced. It is explained how to approximate interval and Sierpinski gasket by sequences of nested self-similar graphs. On these approximating graphs, a sequence of pre-energy forms is defined leading to a limit form on the limit object which turns out to be a Dirichlet form. Moreover, an electric network interpretation of the problem is given, leading to a quite good understanding how the renormalization of the forms has to be done. Finally, the important notion of resistance metric is introduced.
Chapter two is devoted to the Laplacian which can be obtained in a weak form (on a dense domain) from the Dirichlet form via “integration by parts”. Going back to the graph approximation, one also obtains a pointwise formulation. Introducing normal derivatives enables the author to give analogues of the classical Gauss-Green formulae. Moreover, the Green function is introduced and Theorem 2.6.1 provides the existence and uniqueness of solutions of Dirichlet problems. Finally, the local behaviour of harmonic function or, more general, of functions from the domain of the Laplacian is investigated.
The spectrum of the Laplacian is considered in Chapter three. The spectral decimation principle is introduced and applied in order to get the eigenvalues and their multiplicities. Moreover, the existence of localized eigenfunctions is discussed and spectral asymptotics are presented.
In Chapter four, the class of postcritically finite fractals is introduced. Then the ideas and approaches from the first three chapters – where they were developed only for the model cases of interval and Sierpinski gasket – are extended to this class of finitely ramified sets.
The last Chapter collects some miscellaneous topics which are associated with the subject of the book, such as splines and energy measures. Moreover, extensions of the theory to more general classes of fractals as infinite fractals, fractafolds and products of fractals are outlined. Finally, connections with the stochastic “side of the story” such as for example heat kernel estimates are presented.
Each chapter is completed with a large number of notes and references for further reading. Moreover, exercises of different degree of difficulty are provided. The author motivates the presented things very well, and he prefers an informal style which makes the book accessible to anybody who wants to get a quick and short but profound and complete introduction into the topic. From my own experience, it serves well as a textbook – also for undergraduate students.

35-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations
28A80 Fractals
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35P05 General topics in linear spectral theory for PDEs
35R02 PDEs on graphs and networks (ramified or polygonal spaces)