A New Attempt to Construct the Laplace Operator on Fractals

Abstract

One of the most important topics in the analysis of fractals is to construct the Laplacian. But this is actually a particular case of a wider problem – to construct geometrical objects on fractals. Currently, studied methods sometimes lead to difficult problems, require wide knowledge from different branches of mathematics or does not lead to any strict computational methods, which could be easily applied for example in engineering. In this paper, a new attempt is presented. Fractals are treated like objects from so-called differential spaces, i.e. broader category than manifolds. The usefulness of differential spaces is shown in particular fractal situations when one studies some „weird” subsets of n, which are not manifolds themselves.