the biharmonic equation
/ by-har-MON-ik /
Press down on the middle of a thin metal plate clamped at its edges, or a stiff diving board, and watch how it sags. Unlike a flapping string or drumskin, a plate or beam resists bending — it has stiffness, and the way it deflects is governed not by the Laplacian but by applying the Laplacian twice. The biharmonic equation is the master equation of elasticity for bending plates and beams.
It reads Laplacian(Laplacian u) = 0, written Laplacian^2 u = 0 or Delta^2 u = 0, where the unknown u is the deflection of the plate. A function satisfying this is called biharmonic. In two dimensions, written out, it is u_xxxx + 2 u_xxyy + u_yyyy = 0 — a fourth-order PDE, a step up from the second-order Laplace equation. Where Laplace's equation Laplacian u = 0 models membranes and steady heat (governed by surface tension or conduction, which penalize first derivatives), the biharmonic models bending resistance, which penalizes curvature — second derivatives — so the energy involves the square of the Laplacian and minimizing it gives a fourth-order equation. The 1D analogue is the beam (Euler-Bernoulli) equation u_xxxx = load.
Being fourth-order changes the boundary conditions you must supply: a clamped plate fixes both the deflection u and its slope (normal derivative) at the edge — two conditions, as a fourth-order equation demands. The biharmonic equation runs through structural engineering (deflection of floors, bridges, aircraft skins), through slow viscous (Stokes) flow in two dimensions where the stream function is biharmonic, and through continuum mechanics generally. It is the canonical example that not every PDE in the world is second-order — higher-order operators bring richer behaviour and need more data to pin down.
A diving board clamped at the wall: the static deflection u(x) satisfies the beam equation u_xxxx = w (a constant load w), the 1D biharmonic. Clamping means u = 0 and u_x = 0 at the wall (no deflection, no slope); the free tip needs two more conditions on the higher derivatives. Four boundary conditions for a fourth-order equation — exactly as the order demands.
Fourth-order Laplacian^2 u = 0: bending stiffness, needs two conditions per boundary.
Biharmonic functions are not the same as harmonic ones: every harmonic function is biharmonic, but biharmonic functions need not satisfy a maximum principle, so the clean intuitions from Laplace's equation do not all carry over to fourth order.