Laplace's equation and harmonic functions

A function with \( \Delta u = 0 \) is harmonic. Harmonic functions describe equilibria: the steady temperature in a body, the electrostatic potential in empty space, the velocity potential of an ideal flow. In the plane the real and imaginary parts of every holomorphic function are harmonic, which supplies examples: \( x^2 - y^2 = \operatorname{Re} z^2 \), \( x^3 - 3xy^2 = \operatorname{Re} z^3 \) (the example), \( e^x\cos y = \operatorname{Re} e^z \).

Radial solutions come from \( \Delta u = u_{rr} + \frac{n-1}{r}u_r \) in \( \mathbb R^n \). Setting this to zero gives \( u = a + b\ln r \) in the plane and \( u = a + b\,r^{2-n} \) for \( n \ge 3 \), so \( 1/r \) in space. They are harmonic everywhere except the origin, and that singularity is the seed of the fundamental solution two lessons on.

The central fact is the mean value property: if \( u \) is harmonic on a ball, its value at the centre equals its average over every sphere (and every ball) around that centre. Proof: let \( \phi(r) \) be the average of \( u \) over the sphere of radius \( r \). Differentiating under the integral, \( \phi'(r) \) is the average of the outward normal derivative \( \partial_\nu u \) over that sphere, which by the divergence theorem is a multiple of \( \int_{B_r}\Delta u = 0 \). So \( \phi \) is constant, and as \( r \to 0 \) it tends to \( u \) at the centre.

Picture it: a soap film stretched over a bent wire frame. Each point of the film sits at the average height of any small circle around it; the film has no bumps or dips of its own, only the ones the frame forces on it.

Think it: almost everything about harmonic functions follows from averaging. A harmonic function cannot have an interior maximum unless it is constant (the next lesson). It is infinitely differentiable, because averaging against a smooth bump reproduces it. A harmonic function on all of \( \mathbb R^n \) that is bounded is constant (Liouville), because averages over enormous balls centred at two nearby points almost coincide. In fluid mechanics the pressure solves a Poisson equation \( -\Delta p = \partial_i\partial_j(u_iu_j) \), so these facts are where pressure estimates start.