Laplace's equation

Stretch a soap film across a wire loop and it settles into the smoothest possible surface spanning the wire — no bumps, no dimples, every interior point sitting at the average height of its neighbors. Hold the edges of a metal plate at fixed temperatures and wait: the steady interior temperature does the same averaging. The equation that all such equilibrium fields obey is Laplace's equation, the most important elliptic PDE and the source equation behind potential theory.

It is written nabla^2 u = 0, where nabla^2 is the Laplacian (in two dimensions u_xx + u_yy, in three u_xx + u_yy + u_zz). Read it as a statement that the sum of the curvatures in all directions vanishes at every point: wherever the field curves up in one direction it must curve down by exactly as much in another, so no point can be a strict local maximum or minimum. Solutions of Laplace's equation are called harmonic functions, and they have a beautiful mean-value property — the value at any point equals the average of the values on any sphere centered there. This is the precise content of the maximum principle: extremes can only occur on the boundary.

Laplace's equation is the steady-state, source-free heart of physics: the electrostatic potential in a charge-free region, the gravitational potential in empty space, steady heat with no internal sources, the velocity potential of incompressible irrotational flow, and the real and imaginary parts of any analytic function of a complex variable (which is why complex analysis and 2D potential problems are the same subject). Adding a source on the right-hand side turns it into Poisson's equation; setting the source to zero is what makes the problem about pure equilibrium.