Analysis on Fractals

Let \(V_0 = \{q_0, q_1, q_2\} \in \mathbb{R}^2\) and \(F_i(x) = \frac{1}{2}(x+q_i)\) for \(i=0,1,2\). Then \[SG = \overline{\bigcup_{m=1}^{\infty}\bigcup_\limits{|w|=m}F_w(V_0)}\]

We work on the finite graph approximation \(V_m = \bigcup_{|w|=m}F_{w}(V_0)\)

Let \(u: SG \mapsto \mathbb{R}\). Then the Laplacian \(\Delta_{\mu}\) is defined as \[\Delta_{\mu}u(x) = \frac{3}{2}\lim_{m \to \infty}5^m\Delta_mu(x)\]

\(G: SG \times SG \mapsto R\) is called Green’s Function where \[-\Delta u = f, u|_{V_0} = 0 \iff u(x) = \int_{SG}G(x,y)f(y)\,d\mu\]

\(\partial_nu(q_i)\) is the normal derivative of \(u\) at \(q_i\) where \[\partial_nu(q_i) = \lim_{m \to \infty}\Big(\frac{5}{3}\Big)^m(2u(q_i) - u(F^{m}_{i+1}q_i) - u(F^{m}_{i-1}q_i))\]

\(\partial_Tu(q_i)\) is the tangential derivative of \(u\) at \(q_i\) where \[\partial_{T}u(q_i) = \lim_{m\to\infty}5^m(u(F^{m}_{i+1}q_i) - u(F^{m}_{i-1}q_i))\]