Dirichlet problem
Finding a harmonic function on a region that matches prescribed values on the boundary.
The Dirichlet problem on a region asks for a function that is harmonic throughout the region's interior and matches prescribed values on its boundary. It arises directly from steady-state physics -- the interior function might represent equilibrium temperature or electrostatic potential -- and conformal mapping solves it by transferring a hard boundary-value problem on a complicated region to an easy one, typically a disk or a half-plane, where the answer is already known, then mapping that solution back.
Examples
Finding the steady-state temperature inside a disk, given the temperature prescribed all around its boundary circle, is a Dirichlet problem.
A Möbius transformation carrying a half-plane's boundary data to a disk's turns an unfamiliar Dirichlet problem into a familiar one.
See also