---
tag: NTFY23YDEJ
kind: concept
name: Balayage
msc: [31B15, 31C15, 31D05]
sources:
  - url: https://en.wikipedia.org/wiki/Balayage
same_as:
  wikidata: Q4850161
  wikipedia: Balayage
---

## Summary

Balayage, French for "sweeping out", replaces a measure by a measure on a given set.
The new measure has the same potential as the old one outside that set, and on it up to small exceptional sets.
Poincaré introduced it to solve the Dirichlet problem: the charge of a measure in a domain is swept to the boundary through [[harmonic measure|CFD7BPM51W]].
In modern potential theory the balayage of a measure onto any set is defined through the [[reduced function|909B1WYQ4Q]] of its potential.
It is one of the basic operations of classical and axiomatic potential theory.

## Explanations

### L1 human

Think of a point charge inside a hollow metal ball.
Seen from outside the ball, the charge cannot be told apart from a suitable charge spread over the sphere.
Balayage is the rule that produces this spread-out charge.
For a unit charge at the centre of a ball in space, the swept charge is the uniform distribution of total mass $1$ on the sphere.
Its potential agrees with that of the point charge outside the ball and is smaller inside.