Ricardo A. Sáenz

Notes on the eigenfunctions of the Laplacian on the Sierpińsky gasket:  Chapter15.pdf

Notes on the harmonic structure and the Laplacian on the Sierpiński gasket:  Chapter14.pdf

Notes on the Hausdorff dimension and fractal sets:  Chapter13.pdf

Due June 7 Problem 1 For $latex m\ge 2$, there is no Dirichlet eigenfunction on $latex V_m$ with respect to $latex \lambda_m=2$. Problem 2 The Dirichlet eigenfunctions on $latex V_m$ constructed in class, with respect to $latex \lambda_m=5$, are linearly independent. If we add the three eigenfunctions chained from $latex p_1$ to $latex p_2$, from $latex p_2$ to $latex p_3$, and from $latex p_3$ to $latex p_1$, they are linearly dependent. Problem 3 Let $latex \mathcal E_m$ be a sequence of Dirichlet forms such that  $latex \displaystyle \mathcal E_m(u,v) = \sum_{j=1}^N \frac{1}{r_j}\mathcal E_{m-1}(u\circ f_j,v\circ f_j)$ and, given a function $latex u$ on $latex V_0$, $latex \min\{\mathcal E_1(v,v): v|_{V_0} = u\} = \mathcal E_0(u,u)$, then $latex \min\{\mathcal E_m(v,v): v|_{V_{m-1}} = u\} = \mathcal E_{m-1}(u,u)$ for every $latex m\ge1$ and any given function $latex u$ on $latex V_{m-1}$. Problem 4 Calculate all harmonic structures on the interval, s...