the mean-value property

Here is the single most characteristic fact about a harmonic function, and it sounds almost too clean to be true: the value at any point is exactly the average of the values on any circle (or sphere) drawn around that point. No matter how big the circle, as long as it fits inside the region, the centre value equals the rim average. A harmonic function is, at every point and every scale, perfectly self-averaging.

Precisely, if u is harmonic on a region and the ball of radius r centred at a point p lies inside it, then u(p) equals the average of u over the sphere of radius r around p, and also equals the average of u over the solid ball. The reason is the equation itself: Laplacian u = 0 says u has no net curvature, so contributions that would pull the rim average above the centre are exactly cancelled by those that pull it below. Remarkably the converse is also true — a continuous function with the mean-value property on every small sphere is automatically harmonic. So the mean-value property is not just a consequence of Laplace's equation; it is an equivalent definition of harmonicity, and often the more useful one.

Almost the entire qualitative theory falls out of this one property. The maximum principle is immediate: a point where u is largest cannot exceed the average over a surrounding sphere unless u is constant there, so interior maxima are impossible. Smoothness follows because averaging over balls is a smoothing operation. Harnack's inequality, Liouville's theorem, and the convergence theorems all trace back to mean values. It is also the engine behind the probabilistic picture: a harmonic function's value at a point is the expected value of its boundary value reached by a random walk starting there.