Capacity of a set

Remarks

The infimum equals $\int_{\R^n \setminus K} \abs{\nabla u}^2 \, dx$ for the equilibrium potential $u$ of $K$. This function is harmonic off $K$, equal to $1$ on $K$ when $K$ is regular, and tends to $0$ at infinity. Some authors multiply by a constant such as $1 / ((n-2) \sigma_{n-1})$, where $\sigma_{n-1}$ is the area of the unit sphere. Example: for the closed ball $\bar B_r$ of radius $r$, the equilibrium potential is $u(x) = (r / \abs{x})^{n-2}$ for $\abs{x} \ge r$. Then $\abs{\nabla u} = (n-2) r^{n-2} \abs{x}^{1-n}$, and integration in polar coordinates gives $\operatorname{cap}(\bar B_r) = (n-2) \sigma_{n-1} r^{n-2}$. For $n = 3$ this is $4 \pi r$. Non-example of an admissible function: $u = 1/2$ near $K$ does not satisfy $u \ge 1$ on $K$, so it does not enter the infimum. Capacity is monotone, so a point has capacity at most $(n-2) \sigma_{n-1} r^{n-2}$ for every $r > 0$, hence capacity $0$.