---
tag: 6R92RBM72R
kind: definition
name: Mutual energy
msc: [31B15, 31A15]
defines: 6VR8GASZJY
uses: [DYR1QFNGNJ, FQTXV6BQZG, 6TJTWQ5HCR, VDZ5FKYC2T, JHKSF3XFY2, SFSRQCM3EG]
sources:
  - url: https://en.wikipedia.org/wiki/Capacity_of_a_set
---

## Definition

Let $n \ge 2$ and let $\mu$ and $\nu$ be finite positive [[Borel measures|DYR1QFNGNJ]] on $\R^n$ with compact [[supports|FQTXV6BQZG]].
Let $k$ be the [[Newtonian kernel|6TJTWQ5HCR]] $k(x) = \abs{x}^{2-n}$ when $n \ge 3$, and the [[logarithmic kernel|VDZ5FKYC2T]] $k(x) = \log(1/\abs{x})$ when $n = 2$, with $k(0) = +\infty$.
The *mutual energy* of $\mu$ and $\nu$ is
$$I(\mu, \nu) = \iint k(x - y)\, d\mu(x)\, d\nu(y) \in (-\infty, +\infty].$$

## Remarks

The mutual energy is symmetric, $I(\mu, \nu) = I(\nu, \mu)$, and $I(\mu, \mu)$ is the [[energy|SFSRQCM3EG]] $I(\mu)$.
It equals $\int U^\mu \, d\nu$, where $U^\mu$ is the [[potential|JHKSF3XFY2]] of $\mu$.
For a signed measure $\lambda = \mu - \nu$ with $I(\mu)$, $I(\nu)$ and $I(\mu, \nu)$ finite, one puts $I(\lambda) = I(\mu) - 2 I(\mu, \nu) + I(\nu)$.

Example: for Dirac measures $\delta_a$ and $\delta_b$ with $a \ne b$ in $\R^3$, $I(\delta_a, \delta_b) = 1/\abs{a - b}$.
For the uniform probability measure $\sigma$ on the unit sphere in $\R^3$ and $\delta_0$, $I(\sigma, \delta_0) = U^\sigma(0) = 1$.

Non-example: for $\mu = \nu = \delta_0$, the integrand is $k(0) = +\infty$ on the only point of the support, so $I(\mu, \nu) = +\infty$ is not a finite mutual energy.