  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 $\lambda \geq 0$. 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 $\int_{SG} |\Delta f(x)|^2 d \mu (x)$ 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.
