Variational capacity

Summary

Let $K$ be a compact set inside an open set $\Omega \subseteq \R^n$, and let $1 < p < \infty$. The variational capacity, or $p$-capacity, of $K$ is the least $p$-Dirichlet energy $\int_\Omega \abs{\nabla u}^p \, dx$ of a suitable function $u$. Here $u$ is smooth, at least $1$ on $K$, and zero near the boundary of $\Omega$. For $p = 2$ it is the classical condenser capacity. It is the natural measure of size in nonlinear potential theory and for the $p$-Laplace equation. Sets of $p$-capacity zero are the sets that Sobolev functions with $p$-integrable gradient cannot see.