---
tag: TFY9CD3019
kind: definition
name: Capacity of a set
msc: [31B15, 31C15]
defines: 4YZVFGJVSZ
uses: [0R6EWFKRCF]
sources:
  - url: https://en.wikipedia.org/wiki/Capacity_of_a_set
    locator: Harmonic capacity
---

## Definition

Let $n \ge 3$ and let $K \subseteq \R^n$ be a [[compact set|0R6EWFKRCF]].
Let $C_c^\infty(\R^n)$ be the set of smooth real functions on $\R^n$ with compact support.
The *Newtonian capacity* of $K$ is
$$\operatorname{cap}(K) = \inf \set{\int_{\R^n} \abs{\nabla u}^2 \, dx : u \in C_c^\infty(\R^n), \ u \ge 1 \text{ on } K}.$$

## 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 $0$.