Extensions and their Minimizations on the Sierpinski Gasket

Abstract

We study the extension problem on the Sierpinski Gasket ($SG$). In the first part we consider minimizing the functional ${\mathcal{E}}_{\lambda }(f)=\mathcal{E}(f,f)+\lambda \int f^{2}d\mu$ with prescribed values at a finite set of points where $\mathcal{E}$ denotes the energy (the analog of $\int |\nabla f{|}^2$ in Euclidean space) and $\mu$ denotes the standard self-similiar measure on $SG$. We explicitly construct the minimizer $f(x)={\sum }_i{c}_i{G}_{\lambda }(x_i,x)$ for some constants $c_i$, where $G_{\lambda }$ is the resolvent for $\lambda \ge 0$. We minimize the energy over sets in $SG$ by calculating the explicit quadratic form $\mathcal{E}(f)$ of the minimizer $f$. We consider properties of this quadratic form for arbitrary sets and then analyze some specific sets. One such set we consider is the bottom row of a graph approximation of $SG$. We describe both the quadratic form and a discretized form in terms of Haar functions which corresponds to the continuous result established in a previous paper. In the second part, we study a similar problem this time minimizing ${\int }_{SG}|\Delta f(x){|}^{2}d\mu (x)$ for general measures. In both cases, by using standard methods we show the existence and uniqueness to the minimization problem. We then study properties of the unique minimizers.

28 pages, 5 figures