Fractals: from the fractional dimension to the heat equation

The approach to the problem and the results we will recover will follow the same flow of the study of the heat equation of the classical settings: the first step is to find the fundamental solutions to the heat equation, which will give us the tool to obtain a general solution via convolution. After that, the attention goes to the properties of the selected solution; in particular we will show the parabolic maximum principle and the behavior of the solution as t → 0+.

The space Π_∞ consists of harmonic functions of the iterated Laplacian operator Δ^(m) _μ . Since in R all harmonic functions are first degree polynomials, we can identify Π_∞ with the space of functions whose 2m-th derivative is 0, which are precisely the polynomials of degree 2m − 1. In R we know that the polynomials form a dense subset of L²(K), with K being a compact set, by the Stone–Weierstrass theorem. Theorem 2.5.11 shows that not only the polynomials are not dense in the space of L² functions on fractals, but even that they are orthogonal to every eigenfunction. Hence every function can be approximated by polynomials or by eigenfunctions, but not from both. [...]