Harnack's inequality
/ HAR-nahk /
Harmonic functions are not just free of interior extremes — they are remarkably uniform. A positive harmonic function cannot be wildly large at one interior point and tiny at a nearby point: its values across any region safely inside the domain are all comparable, trapped within a fixed ratio of one another. Harnack's inequality is the precise quantitative statement of this togetherness, and it is one of the deepest qualitative facts in potential theory.
Precisely, let u be harmonic and positive on a region. Harnack's inequality says that for any compact subregion K sitting strictly inside, there is a constant C (depending only on K and the domain, not on u) such that the maximum of u over K is at most C times the minimum of u over K. In other words, a positive harmonic function can vary by at most a bounded factor over any region kept away from the boundary — high values and low values are tied together. On a ball of radius R, the explicit version bounds u(x) at distance d from the centre between (R-d)/(R+d) and (R+d)/(R-d) times the centre value u(0).
Its importance is hard to overstate. Harnack's inequality is the tool that converts a one-sided bound (positivity) into a two-sided control, and from it flow strong forms of the maximum principle, Liouville's theorem, compactness theorems for families of harmonic functions, and the regularity theory of far more general elliptic and parabolic equations — Moser and De Giorgi-Nash built the modern theory of equations with rough coefficients precisely by proving a Harnack inequality for them. The one indispensable hypothesis is positivity: drop it and the inequality is false, since a sign-changing harmonic function passes through zero where no fixed ratio can hold.
On the unit disk, a positive harmonic function at radius 1/2 from the centre is squeezed between (1-1/2)/(1+1/2) = 1/3 and (1+1/2)/(1-1/2) = 3 times its centre value. So whatever the boundary data (as long as it is positive), the value halfway out cannot drop below a third of, or rise above triple, the central value.
Positivity plus harmonicity forces all interior values into a fixed ratio band.
Positivity is essential — the inequality compares max to min as a ratio, which only makes sense (and only holds) when u stays positive. The constant C depends on the geometry, blowing up as the subregion K approaches the boundary, which is why the estimate is interior only.