[Paper Review] Laplacian, on the graph of the Weierstrass function

Claire David|arXiv (Cornell University)|Mar 9, 2017

Mathematical Dynamics and Fractals13 references3 citations

TL;DR

This paper constructs a Dirichlet form and a corresponding Laplacian on the graph of the Weierstrass function using a renormalized limit of discrete graph Laplacians, adapting Kigami and Strichartz's framework for analysis on fractals. The key contribution is the rigorous definition of a weak Laplacian on this self-similar, nowhere-differentiable fractal graph, with spectral decimation used to compute its spectrum.

ABSTRACT

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.

Motivation & Objective

• To extend the theory of analysis on fractals—previously limited to self-similar sets like the Sierpiński gasket—to the graph of the Weierstrass function, a continuous, nowhere-differentiable, self-similar curve.
• To define a weak Laplacian on the Weierstrass graph by constructing a Dirichlet form as the renormalized limit of discrete Dirichlet forms on approximating graphs.
• To establish a measure and energy structure compatible with the fractal geometry of the Weierstrass graph, enabling the definition of a well-behaved Laplacian.
• To compute the spectrum of the resulting Laplacian using spectral decimation, mirroring techniques used on the Sierpiński gasket.
• To provide a new framework for studying diffusion and PDEs on irregular, fractal-like domains beyond standard self-similar fractals.

Proposed method

• Adapts Kigami and Strichartz's approach of defining the Laplacian via Dirichlet forms, using a sequence of piecewise linear approximations of the Weierstrass graph.
• Constructs discrete Dirichlet forms on finite graphs approximating the Weierstrass function, with energies defined via differences of function values at adjacent vertices.
• Uses renormalization to scale the discrete Laplacians so that their limit yields a well-defined, continuous operator on the full graph.
• Introduces a specific self-similar measure on the Weierstrass graph, derived from the scaling properties of the function, to weight the Dirichlet energy.
• Applies spectral decimation to compute the spectrum of the Laplacian, solving a recurrence relation derived from the self-similarity of the graph.
• Solves a nonlinear recurrence involving the eigenvalue parameter via a transformation to a conjugate map, leading to a closed-form expression for the spectrum.

Experimental results

Research questions

• RQ1 Can a Laplacian be rigorously defined on the graph of the Weierstrass function, a self-similar, nowhere-differentiable curve?
• RQ2 How can the Dirichlet form and associated energy be constructed on a fractal graph that is not a post-critically finite self-similar set?
• RQ3 What normalization and measure are required to ensure the convergence of discrete graph Laplacians to a well-defined limit operator?
• RQ4 What is the spectral structure of the Laplacian on the Weierstrass graph, and can it be computed via spectral decimation?
• RQ5 How does the geometry of the Weierstrass graph—particularly its complex self-similarity and non-integer dimension—affect the construction of a Laplacian?

Key findings

• A Dirichlet form is constructed on the Weierstrass graph as the renormalized limit of discrete Dirichlet forms on approximating graphs, enabling the definition of a weak Laplacian.
• The Laplacian is well-defined and continuous on the graph of the Weierstrass function under the proposed framework, despite the function's nowhere-differentiability.
• The spectrum of the Laplacian is computed via spectral decimation, yielding a discrete set of eigenvalues that accumulate at infinity.
• The eigenvalue recurrence relation is solved using a conjugate map transformation, leading to a closed-form expression for the spectrum in terms of iterated functions.
• The construction relies on a self-similar measure and a specific energy normalization that respects the fractal geometry of the Weierstrass graph.
• The method generalizes Strichartz’s approach on the Sierpiński gasket to a broader class of fractal graphs, including the Weierstrass function’s graph.