---
tag: ZAT4TV4030
kind: concept
name: Heat potential
msc: [35K08, 35K05, 31C99]
sources:
  - url: https://en.wikipedia.org/wiki/Heat_equation
---

## Summary

A heat potential is the integral of the fundamental solution of the [[heat equation|VP2HN86VAG]] against a measure or a density in space-time.
It plays for the heat equation the role that the Newtonian potential plays for the Laplace equation.
With a density as source, it solves the inhomogeneous heat equation with zero initial data, which is Duhamel's principle.
Heat potentials are the basic objects of the potential theory of the heat equation, with its own notions of capacity and of regular boundary points.

## Explanations

### L1 human

Drop a unit of heat at one point at one moment: the temperature afterwards is the [[heat kernel|KD404MF042]], a Gaussian bump that spreads out as time passes.
If heat is added at many points and many moments, the temperatures from all the drops add up.
The heat potential is this sum, written as an integral.
Unlike the Newtonian potential, it only looks into the past: heat added later cannot affect the temperature now.