the Dirichlet problem
/ DEER-ish-lay or dee-ree-KLAY /
Suppose you can control the temperature all along the edge of a plate — you clamp each boundary point to a chosen value — and you ask what the temperature will be inside once it settles. That question is the Dirichlet problem: given the boundary values, find the harmonic function inside that matches them. It is the most natural and most studied boundary-value problem for Laplace's and Poisson's equations.
Precisely, on a region D with boundary, the Dirichlet problem for Laplace's equation asks for a function u with Laplacian u = 0 inside D and u = g on the boundary, where g is a prescribed boundary function. The Poisson version replaces the interior equation with Laplacian u = f. The two pillars of the theory are that the solution is unique — two solutions with the same boundary data would have a harmonic difference that is zero on the boundary, and the maximum principle then squeezes it to zero everywhere — and that, for reasonable domains and data, a solution exists. On a disk it is given explicitly by the Poisson integral formula; on a rectangle by a Fourier series; on general domains existence is secured by Dirichlet's principle (energy minimization) or by the Perron method of subharmonic functions.
Existence is more delicate than uniqueness, and honesty matters here. For very rough domains the Dirichlet problem can fail to have a solution that takes the boundary values continuously — Lebesgue's spine is the classic counterexample, a domain with an inward cusp so sharp that a boundary point is irregular. Which boundary points are well-behaved is captured by the notion of a barrier and, more quantitatively, by capacity. For smooth domains, though, the Dirichlet problem is the well-posed, solvable cornerstone of potential theory.
Heat the top edge of a square to 100 degrees, keep the other three at 0, and ask for the steady interior temperature. The answer is a Fourier sine series in x whose coefficients decay, each term multiplied by a hyperbolic profile in y that grows from the cold sides toward the hot edge — a fully explicit harmonic solution matching the prescribed boundary data.
Prescribe the values on the whole boundary; the unique harmonic interior follows.
It is the boundary VALUES that are prescribed, not the boundary flux — that latter problem is the Neumann problem. And do not confuse this with an initial-value problem: Laplace's equation is elliptic, so data go on the whole boundary at once, not on an initial slice that you march forward in time.