Dirichlet Problem - Example: The Unit Disk in Two Dimensions

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 R2 is given by the Poisson integral formula.

If is a continuous function on the boundary of the open unit disk, then the solution to the Dirichlet problem is given by

u(z) = \begin{cases} \frac{1}{2\pi}\int_0^{2\pi} f(e^{i\psi}) \frac {1-\vert z \vert ^2}{\vert 1-ze^{-i\psi}\vert ^2} d \psi & \mbox{if }z \in D \\ f(z) & \mbox{if }z \in \partial D. \end{cases}

The solution is continuous on the closed unit disk and harmonic on

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

where is harmonic

and chosen such that for .

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