---
tag: EG8Z3FWV9A
kind: concept
name: Double layer potential
msc: [31B10, 31A10]
sources:
  - url: https://en.wikipedia.org/wiki/Double_layer_potential
same_as:
  wikidata: Q5300057
  wikipedia: Double layer potential
---

## Summary

A double layer potential is the potential of a layer of dipoles spread over a surface, with dipoles pointing along the normal.
It is the integral of the normal derivative of the [[fundamental solution of the Laplacian|BSJ8X2XRN9]] against a density on the boundary of a domain.
It is [[harmonic|0DEB0JMN0W]] off the surface, and its value jumps across the surface by exactly the density, as the [[jump relations|J58YA8QNNX]] say.
This jump makes it the standard tool for turning the [[Dirichlet problem|8D56GQ4AG8]] into a Fredholm integral equation, see the [[boundary integral method|QQ62R5ZAXJ]].

## Explanations

### L1 human

Put a thin sheet of tiny magnets on a surface, each with its north pole just outside and its south pole just inside.
The potential they create is a double layer potential.
When you cross the sheet, the potential jumps up by the strength of the magnets at that place.
For the unit sphere in space with constant strength $1$, the potential is $-1$ inside the sphere and $0$ outside.