Harnack's inequality
/ HAR-nahk /
Take a positive harmonic function — a steady temperature that is warm everywhere, never reaching zero. Harnack's inequality says that across any region sitting safely inside the domain, the function cannot be wildly uneven: its biggest value there is at most some fixed multiple of its smallest value. A positive harmonic function that is large somewhere on a compact piece is large everywhere on that piece. It cannot be huge at one point and almost zero a short distance away.
Precisely: if u >= 0 is harmonic on a region, and K is a compact subset sitting strictly inside, then 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. So sup_K u <= C inf_K u. The constant comes from geometry alone; the same C works for every positive harmonic function. For the unit ball there is even an explicit version from the Poisson integral formula, the classic Harnack inequality, that bounds u at an interior point by u at the centre times an explicit factor depending only on the distance to the boundary. There is a parabolic Harnack inequality too, but it has a twist: comparing values at different points requires also waiting a little time, because diffusion needs time to even out — the parabolic version compares the present at one point with a slightly later time at another.
Why is this powerful? Because it controls oscillation, and controlling oscillation is the gateway to regularity. The celebrated De Giorgi-Nash-Moser theorem proves that solutions of elliptic equations with merely bounded (not smooth) coefficients are automatically Holder continuous, and the heart of Moser's proof is exactly a Harnack inequality. It also gives Liouville's theorem in one stroke (a positive harmonic function on all of space must be constant) and convergence results for sequences of positive solutions.
If a positive harmonic temperature on a region reads 4 degrees at one interior point, then at any nearby interior point inside the same compact core it must read at least 4/C and at most 4C degrees, where C is fixed by geometry. Knowing the value at one point pins down the size at all of them.
On a compact interior set, the max and min of a positive solution differ by at most a geometric constant.
Positivity is essential — Harnack says nothing about sign-changing solutions, since you cannot bound a max by a min that might be zero or negative. And in the parabolic case the inequality is one-directional in time: you compare an earlier value at one point with a later value at another, never the reverse.