Capacity of a set

Summary

The Newtonian capacity of a compact set $K \subseteq \R^n$, $n \ge 3$, measures how much electric charge $K$ can hold at unit potential. It is the least Dirichlet energy of a function that is at least $1$ on $K$ and decays at infinity. Capacity is the right notion of size in potential theory: sets of capacity zero are the negligible sets for harmonic functions and for Sobolev functions. In the plane its role is played by logarithmic capacity.