Editing Dirichlet problem (section)

==Example: the unit disk in two dimensions==
In some simple cases the Dirichlet problem can be solved explicitly. For example, the solution to the Dirichlet problem for the unit disk in '''R'''<sup>2</sup> is given by the [[Poisson integral formula]].<ref>https://www.diva-portal.org/smash/get/diva2:1748016/FULLTEXT01.pdf</ref>

If <math>f</math> is a continuous function on the boundary <math>\partial D</math> of the open unit disk <math>D</math>, then the solution to the Dirichlet problem is <math>u(z)</math> given by

: <math>u(z) =
 \begin{cases}
  \displaystyle \frac{1}{2\pi} \int_0^{2\pi} f(e^{i\psi}) \frac {1 - |z|^2}{|1 - ze^{-i\psi}|^2} \,d\psi & \text{if } z \in D, \\
  f(z) & \text{if } z \in \partial D.
 \end{cases}
</math>

The solution <math>u</math> is continuous on the closed unit disk <math>\bar{D}</math> and harmonic on <math>D.</math>

The integrand is known as the [[Poisson kernel]]; this solution follows from the Green's function in two dimensions:

: <math>G(z, x) = -\frac{1}{2\pi} \log|z - x| + \gamma(z, x),</math>

where <math>\gamma(z, x)</math> is [[Harmonic function|harmonic]] (<math>\Delta_x \gamma(z, x) = 0</math>) and chosen such that <math>G(z, x) = 0</math> for <math>x \in \partial D</math>.