Fractals: from the fractional dimension to the heat equation

In this thesis we will introduce the world of fractals and explore some of their properties and capabilities. These structures can be characterized in different ways, but the main feature for which they are known is that many of them have a "fractional dimension", something that defies the usual perception that a line has dimension one and the plane has dimension two.

In Chapter 1, we will understand different notions of dimension and we will define a fractal as a set that behaves in a specific way with respect to some of these definitions; in particular, we will state some results characterizing these sets in terms of their dimension. After that, we will conclude the chapter by showing a graph theoretical approach to fractals that will enable us to give a closed formula for the dimension of the fractal.

In Chapter 2, we will deepen the analysis of fractals with the scope of studying the heat equation on a fractal domain. The first obstacle to this program is to define correctly the differential operators on fractal sets: we will overcome it by taking the limit of a suitable difference operator on an "increasing" sequence of resistance networks. Under suitable assumptions, the sequence of resistance networks will stem naturally from the so-called "harmonic structure" associated with the fractal and it will converge to it.

Similarly to what happens in the modern theory of partial differential equations, it will be appropriate to define the Sobolev spaces H1 (Ω) and H1 0 (Ω) on a fractal set Ω in order to encode the boundary conditions within the operator itself. The definition of these spaces is not immediate and is related to the need of a more general notion of the Laplacian operator than the usual one on regular domains. The definition of Laplacian operator that we will provide enjoys many interesting properties, among which some unique spectral properties, such as possessing eigenfunctions that satisfy both Neumann and Dirichlet boundary conditions.

Although it is possible to study partial differential equations on fractals with all type of boundary condition, for the scope of this thesis, we will only cover Dirichlet- and Neumann-type boundary conditions. The chapter will culminate with the study of the heat equation ∂tu − ∆u = 0 with assigned starting condition u0 ∈ L2 (K, µ) , where we will give the solution in terms of convolution of the initial datum u0 with a specific function that will be referred to as the heat kernel (recall that in the classical framework it is the fundamental solution to the heat equation ∂tu − ∆u = δ0).

Two appendices complement this work. The first one may be interesting for the curious reader that wants to see some exotic examples of fractals with astonishing properties such as curves that fill the two-dimensional space or the famous Mandelbröt set, that was inspirational to many mathematicians to progress in the study of fractal sets.
The second appendix offers a different, more abstract perspective to the material presented in Chapter 2. Indeed, it contains some results that the author deems interesting and inspiring to be included in this thesis, without diverting the attention from the main discussion in Chapter 2.