Condenser capacity

Summary

A condenser is a pair $(K, \Omega)$ of a compact set $K$ inside an open set $\Omega \subseteq \R^n$. Its capacity is the least Dirichlet energy $\int_\Omega \abs{\nabla u}^2 \, dx$ of a function that is at least $1$ on $K$ and vanishes near the boundary of $\Omega$. It is the case $p = 2$ of the variational capacity. Condenser capacity works in every dimension, including the plane. It is a basic tool in the theory of quasiconformal maps through the conformal modulus of ring domains.