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Extensions and their Minimizations on the Sierpinski Gasket

We study the extension problem on the Sierpinski Gasket ($SG$). In the first part we consider minimizing the functional $\mathcal{E}_λ(f) = \mathcal{E}(f,f) + λ\int f^2 d μ$ 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 $μ$ denotes the standard self-similiar measure on $SG$. We explicitly construct the minimizer $f(x) = \sum_{i} c_i G_λ(x_i, x)$ for some constants $c_i$, where $G_λ$ is the resolvent for $λ\geq 0$. 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 $\int_{SG} |Δf(x)|^2 d μ(x)$ 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.