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 μ 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 _λ is the resolvent for λ≥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.