the minimal-surface equation
Dip a bent wire loop into soapy water and pull it out: the film that spans it is, of all the surfaces that could fill the loop, the one with the least area. Nature solves an optimization problem instantly. The minimal-surface equation is the PDE that this least-area surface must satisfy — the precise mathematical statement of 'no nearby surface with the same boundary has smaller area'.
Write the surface as a graph z = u(x, y) over a region in the plane. Its area is the integral of sqrt(1 + |grad u|^2). Minimizing that area functional and computing its Euler-Lagrange equation gives the minimal-surface equation: div( grad u / sqrt(1 + |grad u|^2) ) = 0. Unpacked, this says the mean curvature of the graph is zero everywhere — the surface curves up in one direction exactly as much as it curves down in the perpendicular direction, so a soap film is saddle-shaped, never dome-shaped (a dome has nonzero mean curvature and could shrink its area). It is a quasilinear elliptic equation: the principal coefficients depend on grad u through that square-root denominator. For nearly-flat surfaces (|grad u| small) the denominator is about 1 and the equation is approximately Laplace's equation Laplacian u = 0 — minimal surfaces are the genuinely nonlinear cousins of harmonic functions, the two coinciding in the small-slope limit.
Why does it matter? It is the founding example of the calculus of variations and of geometric PDE, the prototype for the whole study of how geometry minimizes area or energy. Its solutions have famous rigidity: Bernstein's theorem says an entire minimal graph over the whole plane must be a flat plane (a striking uniqueness result that fails in high enough dimension — honesty: it holds up to dimension 8 and then breaks, one of geometry's great surprises). And it is the static, equilibrium picture whose time-dependent partner is mean-curvature flow: a non-minimal surface evolving to reduce its area moves by mean-curvature flow and, if it settles, settles onto a minimal surface.
The catenoid (the surface you get by spinning a hanging-chain curve about an axis) and the helicoid (a spiral ramp) are classic minimal surfaces — soap films spanning the right wire frames. Both satisfy div(grad u / sqrt(1 + |grad u|^2)) = 0, and both are saddle-like at every point, curving up and down equally, so their mean curvature is exactly zero everywhere.
Zero mean curvature: the soap film that minimizes area for its boundary.
'Minimal' means stationary for area (zero mean curvature), not necessarily globally least-area — a minimal surface is a critical point of area, and like any critical point it can be a saddle rather than a true minimum. Also, it reduces to Laplace's equation ONLY in the small-slope approximation; in general it is genuinely nonlinear.