Harnack's inequality
/ HAR-nahk /
A positive harmonic function — think of a temperature distribution that is everywhere above some baseline, or a potential that never goes negative — cannot wander wildly. Harnack's inequality is the precise statement of this tameness: on any region sitting safely inside the domain, the largest value of a positive harmonic function is controlled by a fixed multiple of its smallest value. Positivity alone forbids the function from being huge in one spot and tiny in a neighboring spot.
On the unit disk a clean version comes straight from the Poisson kernel. If u is harmonic and positive on the disk and z = r e^(i theta) with |z| = r, then ((1 - r) / (1 + r)) u(0) <= u(z) <= ((1 + r) / (1 - r)) u(0). The two factors are exactly the smallest and largest values of the Poisson kernel at distance r, so the value at z is trapped between two constant multiples of the central value u(0). More generally, on any compact set K contained in a region D there is a constant C, depending only on K and D (not on u), such that max over K of u is at most C times min over K of u, for every positive harmonic u on D.
Harnack's inequality is a workhorse of potential theory. Its main use is controlling limits: it shows that an increasing sequence of harmonic functions cannot blow up in a patchy, ill-behaved way — if it is bounded somewhere it is controlled everywhere on compact sets — which is the heart of Harnack's principle and of the Perron method for the Dirichlet problem. The essential hypothesis to respect: positivity (or a uniform lower bound). Drop it and the inequality is false — a harmonic function that changes sign, like u = x, has min zero on a circle through the origin while its max there is positive, so no finite multiple relates them.
For a positive harmonic u on the unit disk, at radius r = 1/2 Harnack gives (1/2 / (3/2)) u(0) <= u(z) <= (3/2 / (1/2)) u(0), that is (1/3) u(0) <= u(z) <= 3 u(0): the value anywhere on the circle |z| = 1/2 stays between one-third and three times the center value.
On the circle of radius 1/2, a positive harmonic function is pinned between u(0)/3 and 3 u(0).
The constant blows up as r -> 1 (the factor (1 + r)/(1 - r) -> infinity), so Harnack controls behavior only on sets bounded away from the boundary — it says nothing about how a positive harmonic function may grow as you approach the edge.