The group of papers collected in this volume covers several aspects of fractal theory.
Fractals can be seen as intrinsic manifolds of peculiar topological and spectral nature. This is the point of view taken in the paper by Allan, Barany and Strichartz. This paper is a theoretical and computational contribution to the study of the Laplace operator, and of some operator functions of the Laplacian, on the important fractal known as the Sierpinski gasket.
Fractals can also be seen as closed subsets of a Euclidean space. The trace theory for function spaces becomes particularly challenging when the traces are taken on an irregular closed set of fractal type. Due to the lack of differentiability of the sets, the trace function spaces are defined by means of atomic decompositions. The first paper by Jonsson is a detailed description of the construction of Besov spaces on closed sets. The second paper by Jonsson investigates the class of compact sets which preserve Markov's inequality.
In the paper by Hinz and Zähle, fractals occur as stochastic perturbation of certain systems of partial differential equations, which are allowed to be possibly of fractional Brownian type. This paper poses the foundations for the study that the authors are carrying on at the confluence of analysis and probability.
Fractals may also take the form of complicated measures and functions of inhomogeneous type, known as multifractals. While a homogeneous fractal, like the Sierpinski gasket, has the same dimension at every point, multifractals give rise to a full spectrum of dimensions varying in the structure. A powerful tool of analysis of these dimensional features is provided by certain complex zeta functions, whose poles take the role of complex dimensions. New ideas and perspectives in this fascinating subject are illustrated in the paper by Lapidus and Rock.
As a whole, this group of selected papers shows how rich and multifaceted the current research on fractal theories is and how promising the new developments are.
We wish to thank all the authors who contributed their works. We hope that this special issue of CVEE will help in disseminating their ideas and results further and in gaining new readers to this interesting new field of analysis.