  The Laplacian plays a major role in the mathematical analysis of partial
differential equations. Recently, the work of J. Kigami, taken up by R. S.
Strichartz, allowed the construction of an operator of the same nature, defined
locally, on graphs having a fractal character: the triangle of Sierpinski, the
carpet of Sierpinski, the diamond fractal, the Julia sets, the fern of
Barnsley.
  Strangely, the case of the graph of the Weierstrass function, introduced in
1872 by K. Weierstrass, which presents self similarity properties, does not
seem to have been considered anywhere. It is yet an obligatory passage, in the
perspective of studying diffusion phenomena in irregular structures.
  We have asked ourselves the following question: given a continuous function u
on the graph of the Weierstrass function, under which conditions is it possible
to associate to u a function Delta u which is, in the weak sense, its Laplacian
?
  We present, in the following, the results obtained by following the approach
of J. Kigami and R. S. Strichartz. Ours is made in a completely renewed
framework, as regards, the one, affine, of the Sierpinski gasket. First, we
concentrate on Dirichlet forms, on the graph of the Weierstrass function, which
enable us the, subject to its existence, to define the Laplacian of a
continuous function on this graph. This Laplacian appears as the renormalized
limit of a sequence of discrete Laplacians on a sequence of graphs which
converge to the one of the Weierstrass function. The normalization constants
related to each graph Laplacian are obtained thanks Dirichlet forms. The
spectrum of the Laplacian thus built is obtained through spectral decimation.
