David Simmons

Abstract: In this talk, we introduce the capacity of a set in \$\mathbb{R}^n\$. This is another notion of the size of a set and is a mathematical generalization of how much electrical charge the set can hold. After looking at basic properties, we'll explore how capacity relates to Hausdorff measure. Ultimately, we'll prove that Ahlfors regularity provides a lower bound for the density of the capacity. For example, this is a sufficient condition for the boundary of a domain to satisfy Wiener's criterion, a characterization of the regularity of the solution to the Dirichlet problem at the boundary.