Laplace's equation
/ lah-PLAHSS /
Imagine a thin metal plate whose edges are held at fixed temperatures — one side warm, one side cold — and you wait a long time until nothing changes any more. The temperature in the interior has settled into a smooth, balanced pattern: every point is exactly the average of its neighbours, with no hot or cold spots forming on their own. Laplace's equation is the rule that describes this state of perfect equilibrium. It is the most famous of all elliptic equations.
Written out, the equation is Laplacian u = 0, where the Laplacian is the sum of the unmixed second derivatives: in two dimensions u_xx + u_yy = 0, in three dimensions u_xx + u_yy + u_zz = 0. The Laplacian of u at a point measures how much u there differs from its average over a tiny surrounding sphere. Setting it to zero says there is no such difference anywhere — u sits exactly at its local average everywhere. A solution u is called a harmonic function. Because there is no time variable, u depends only on position: Laplace's equation governs steady states, not evolution.
It appears everywhere a quantity is in balance: the steady-state temperature once the heat equation has stopped changing (u_t = 0), the electrostatic potential in a charge-free region, the gravitational potential in empty space, the velocity potential of an incompressible, irrotational fluid. Because it is elliptic and has no characteristics, it is smoothing and rigid: a solution is determined by its values on the whole boundary, and prescribing Cauchy data on a curve (value and normal derivative) is famously ill-posed (Hadamard). It is the opposite of the wave equation in temperament — all equilibrium, no propagation.
On a square plate with three sides held at 0 degrees and the top edge held at 100 degrees, the interior temperature is the harmonic function matching those boundary values; it is smooth inside, equals 100 only on the hot edge, and never rises above 100 or below 0 anywhere inside.
The boundary data alone pins down the unique steady state — and the maximum principle forbids any interior extreme.
Laplace's equation is elliptic, so it has no real characteristic curves and is not solved as an initial-value problem the way the wave equation is. Posing it as a Cauchy problem (data on one curve) is ill-posed; the natural, well-posed problems are the Dirichlet and Neumann boundary-value problems.