A note on energy forms on fractal domains

Abstract

J. Kigami has laid the foundations of what is now known as analysis on fractals, by allowing the construction of an operator of the same nature of the Laplacian, defined locally, on graphs having a fractal character. The Sierpinski gasket stands out of the best known example. It has, since then, been taken up, developed and popularized by R. S. Strichartz. The Laplacian is obtained through a weak formulation, obtained by means of Dirichlet forms, built by induction on a sequence of graphs that converges towards the considered domain. It is these Dirichlet forms that enable one to obtain energy forms on this domain. Yet, things are not that simple. If, for domains like the Sierpinski gasket, the Laplacian is obtained in a quite natural way, one must bear in mind that Dirichlet forms solely depend on the topology of the domain, and not of its geometry. Which means that, if one aims at building a Laplacian on a fractal domain, the topology of which is the same as, for instance, a line segment, one has to find a way of taking account a very specific geometry. We came across that problem in our work on the graph of the Weierstrass function The solution was thus to consider energy forms more sophisticated than classical ones, by means of normalization constants that could, not only bear the topology, but, also, the very specific geometry of, from now on, we will call W-curves. We presently aim at investigating the links between energy forms and geometry. We have chosen to consider fractal curves, specifically, the Sierpi\'nski arrowhead curve, the limit of which is the Sierpi\'nski gasket. Does one obtain the same Laplacian as for the triangle ? The question appears as worth to be investigated.