the Dirichlet problem

Suppose you can paint a temperature pattern only on the rim of a metal plate — say hot on the right half of the edge, cold on the left — and then let the plate settle. What does the temperature look like everywhere inside once it stops changing? Asking for that interior pattern, given the prescribed values on the boundary, is the Dirichlet problem. It is the central boundary-value problem of potential theory.

Stated cleanly: given a region D and a continuous function g defined on its boundary, find a function u that is harmonic inside D (so u_xx + u_yy = 0) and that matches g on the boundary, meaning u extends continuously to the boundary with u = g there. Two questions come bundled in: does such a u exist, and is it unique? Uniqueness is easy and follows from the maximum principle — two solutions with the same boundary data would have a harmonic difference that is zero on the boundary, hence zero everywhere. Existence is the harder half: it holds for the disk by an explicit formula (the Poisson integral), for nice regions by transplanting the disk's answer through a conformal map, and for surprisingly general regions by the Perron method using subharmonic functions and barriers.

The Dirichlet problem is where harmonic-function theory earns its keep. Solving it tells you the steady temperature, the electrostatic potential with prescribed boundary voltage, or the equilibrium shape of a membrane pinned along a wire — all the same mathematics. The honest caveat: not every region with every boundary behaves perfectly. At certain bad boundary points (think the tip of an inward spike, or an isolated puncture) the solution may fail to take the prescribed boundary value continuously; such points are called irregular, and characterizing the regular ones (those equipped with a barrier) is a genuine part of the theory.