Extensions and their Minimizations on the Sierpinski Gasket

TL;DR

This paper investigates the extension and minimization problems on the Sierpinski Gasket, explicitly constructing solutions and analyzing their properties for energy functionals and Laplacian-based functionals.

Contribution

It provides explicit constructions of minimizers for energy functionals on the Sierpinski Gasket and analyzes their properties, including discretized forms and applications to specific sets.

Findings

Explicit minimizer formulas using resolvent functions

Analysis of quadratic forms for arbitrary and specific sets

Existence and uniqueness results for minimizers

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 $\lambda \geq 0$. 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.