# American Institute of Mathematical Sciences

January  2011, 10(1): 1-44. doi: 10.3934/cpaa.2011.10.1

## Analysis of the Laplacian and spectral operators on the Vicsek set

 1 Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton NJ 08544-1000, USA Government 2 Mathematics Department, Malott Hall, Cornell Univeristy, Ithaca, NY 14853, United States 3 Department of Mathematics, Brown University, Box 1917, Providence, RI 02912, USA Government

Received  January 2010 Revised  April 2010 Published  November 2010

We study the spectral decomposition of the Laplacian on a family of fractals $\mathcal{VS}_n$ that includes the Vicsek set for $n=2$, extending earlier research on the Sierpinski Gasket. We implement an algorithm [23] for spectral decimation of eigenfunctions of the Laplacian, and explicitly compute these eigenfunctions and some of their properties. We give an algorithm for computing inner products of eigenfunctions. We explicitly compute solutions to the heat equation and wave equation for Neumann boundary conditions. We study gaps in the ratios of eigenvalues and eigenvalue clusters. We give an explicit formula for the Green's function on $\mathcal{VS}_n$. Finally, we explain how the spectrum of the Laplacian on $\mathcal{VS}_n$ converges as $n \to \infty$ to the spectrum of the Laplacian on two crossed lines (the limit of the sets $\mathcal{VS}_n$.)
Citation: Sarah Constantin, Robert S. Strichartz, Miles Wheeler. Analysis of the Laplacian and spectral operators on the Vicsek set. Communications on Pure and Applied Analysis, 2011, 10 (1) : 1-44. doi: 10.3934/cpaa.2011.10.1
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