Harnack's principle
/ HAR-nahk /
Stack up harmonic functions in an increasing order — each one at least as large as the one before, everywhere — and ask what their limit looks like. Harnack's principle gives a clean dichotomy: either the limit shoots off to infinity uniformly on every compact piece, or it settles down to a perfectly good harmonic function. There is no middle ground, no ragged limit that is finite here and infinite there in a tangled way.
The precise statement: let u_1 <= u_2 <= u_3 <= ... be an increasing sequence of harmonic functions on a connected region D. Then exactly one of two things happens. Either u_n(z) -> +infinity uniformly on every compact subset of D, or the u_n converge (uniformly on compact subsets) to a limit function u that is itself harmonic on D. The mechanism is Harnack's inequality applied to the nonnegative differences u_n - u_m: positivity forces the sequence to grow or converge in lockstep across any compact set, so convergence at a single point spreads to convergence everywhere, and the uniform-on-compacta limit of harmonic functions is harmonic.
This is the precise tool that makes the Perron method for the Dirichlet problem work: there one builds the solution as the supremum (a kind of increasing limit) of a family of subharmonic functions, and Harnack's principle is what guarantees the resulting candidate is genuinely harmonic rather than merely some limit with no structure. It is the harmonic-function analogue of the theorems (Weierstrass, Montel) that keep limits of holomorphic functions holomorphic. A point worth stressing: monotonicity is essential — without the increasing (or decreasing) hypothesis, a limit of harmonic functions need not be harmonic, and the clean either-or dichotomy collapses.
Let u_n(z) = (1 - 1/n) times the harmonic function Re((1 + z)/(1 - z)), the Poisson kernel's potential; each u_n is harmonic and positive, the sequence increases to its limit, and by Harnack's principle that limit is again harmonic on the disk.
An increasing sequence of harmonic functions whose limit Harnack's principle certifies as harmonic.
The dichotomy is between locally-uniform convergence to a harmonic function and locally-uniform divergence to +infinity; the increasing sequence cannot converge to something non-harmonic, but if you drop monotonicity all bets are off.