---
tag: 8D56GQ4AG8
kind: problem
name: Dirichlet problem
msc: [31A25, 35J25, 31B20]
uses: [0DEB0JMN0W, 0EWMDD6C14, 05D5EGMFYV, 0X3WGX95FE, 0E0B6RA8K0]
about: [0DEB0JMN0W]
sources:
  - url: https://en.wikipedia.org/wiki/Dirichlet_problem
same_as:
  wikidata: Q1192869
  wikipedia: Dirichlet problem
history:
  posed_by: "Peter Gustav Lejeune Dirichlet"
---

## Problem

Let $\Omega$ be a bounded [[domain|0EWMDD6C14]] in $\R^n$, for example in $\C = \R^2$, with [[boundary|05D5EGMFYV]] $\partial\Omega$.
Let $\varphi \colon \partial\Omega \to \R$ be a [[continuous function|0X3WGX95FE]].
Find a continuous function $u \colon \overline{\Omega} \to \R$ that is [[harmonic|0DEB0JMN0W]] in $\Omega$, that is, its [[Laplacian|0E0B6RA8K0]] $\Delta u$ vanishes there, with $u = \varphi$ on $\partial\Omega$.

## Remarks

A solution, if it exists, is unique, by the maximum principle for harmonic functions.
For the unit disc, the solution exists for every $\varphi$ and is the Poisson integral of $\varphi$ against the [[Poisson kernel|TCYSWCTYYM]].
For a general bounded domain, Perron's method produces a candidate solution as the upper envelope of [[subharmonic functions|Z1MHMGFNFW]] lying below $\varphi$ on the boundary.
The candidate takes the boundary value $\varphi$ at every regular boundary point.
The problem is solvable for every continuous $\varphi$ if and only if every boundary point is regular.
Every boundary point of a planar domain bounded by finitely many Jordan curves is regular.
Example of failure: let $\Omega = \set{z \in \C : 0 < \abs{z} < 1}$, with $\varphi = 0$ on $\abs{z} = 1$ and $\varphi(0) = 1$.
A solution would be bounded and harmonic near $0$, so it would extend to a harmonic function on the whole disc.
This function has boundary values $0$, so it is $0$ everywhere, which contradicts $u(0) = 1$.

## Explanations

### L1 human

Fix the temperature along the edge of a metal plate and wait until the temperature stops changing.
The Dirichlet problem asks for the temperature inside the plate.
The equilibrium temperature is a harmonic function, so the problem is to find a harmonic function with given values on the boundary.
For nice domains there is always exactly one answer, but for domains with very thin pieces of boundary, such as an isolated point, there may be none.