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Analysis on fractals. (English) Zbl 0998.28004
Cambridge Tracts in Mathematics. 143. Cambridge: Cambridge University Press. viii, 226 p. (2001).
The book is based on a graduate course at Cornell University in 1997. Thus, it is the most recent introduction to the analysis of “Laplacians” on what physicists call finitely ramified self-similar fractals. It is a nice collection of results previously scattered in a vast number of articles complemented by several new results. Every chapter closes with a collection of exercises. A bibliography of 187 entries provides access to the fast growing area of analysis on fractals.
Mathematically speaking Kigami’s class of post critically finite self-similar fractals is considered and their Dirichlet forms, Dirichlet operators, spectral properties and heat semigroups are studied analytically. Only Aronson type estimates of heat kernels are treated probabilistically because there is no analytic argument so far. The central part of the book is Chapter 3 on the construction of Dirichlet forms, harmonic functions, Green’s functions and “Laplacians” on suitable fractals of the above kind.
Chapter 1 treats the geometry of self-similar sets along the lines of Hutchinson, Hata, Kigami and Moran. Some results on connected fractals are new.
Chapter 2 is about the analysis on limits of networks. Symmetric Dirichlet forms on finite sets are reinterpreted as electrical resistor networks. The convergence of a “compatible” sequence of Dirchlet forms on a sequence of increasing sets is described. Here “compatible” means that the effective resistance of a pair of points in a given set does not change with increasing sequence index.
In Chapter 3 on the construction of Laplacians on p.c.f. self-similar structures a local regular Dirichlet form is constructed on the fractal starting from a “compatible” sequence of discrete Dirichlet forms. Newcomers to the field should think of discrete approximations to the one dimensional classical Laplacian on the open unit interval to see the broad line of arguments. The resulting harmonic structure “lives” on a subset of the fractal and equals the fractal if and only if the harmonic structure is regular. New is the study of the non-regular case.
Chapter 4 about the eigenvalues and eigenfunctions of Laplacians shows a Weyl-type estimate for the asymptotic frequency of eigenvalues and the existence of localized eigenfunctions. Both observations possess typical fractal aspects as opposed to classical Laplacians. A subsection on estimates of eigenfunction is new.
Chapter 5 on heat kernels considers Neumann and Dirichlet boundary conditions while estimating the density of the corresponding heat semigroups.

MSC:
28A80 Fractals
31C25 Dirichlet forms
28-02 Research exposition (monographs, survey articles) pertaining to measure and integration
31-02 Research exposition (monographs, survey articles) pertaining to potential theory
31C20 Discrete potential theory
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35K05 Heat equation
58J35 Heat and other parabolic equation methods for PDEs on manifolds
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