Liouville's theorem for harmonic functions
/ lyoo-VEEL /
How much room does a harmonic function have to roam if it must stay tame on all of infinite space? Almost none. Liouville's theorem says that a harmonic function defined on the entire plane (or all of space) that stays bounded must be a constant. There is no bounded, non-constant equilibrium spread over the whole universe — the self-averaging rigidity of harmonicity, applied at every scale out to infinity, flattens everything to a single value.
Precisely, if u is harmonic on all of n-dimensional space and bounded (its values stay between some fixed limits), then u is constant. The cleanest proof uses the mean-value property together with the gradient estimate it implies: the gradient of u at any point is controlled by the average of u over a huge ball around it, and as the ball grows to infinity that control forces the gradient to vanish everywhere. A zero gradient everywhere means u never changes — it is constant. Sharper versions allow growth: a harmonic function on all of space that grows no faster than a polynomial of degree k must itself be a polynomial of degree at most k, so bounded (k = 0) gives a constant.
The theorem mirrors its famous cousin in complex analysis — a bounded entire analytic function is constant — which is no coincidence, since the real part of an entire function is harmonic on the whole plane. Its uses are leverage-style: many existence and uniqueness arguments blow a local solution up to a global one, show it is bounded and harmonic on all of space, and conclude by Liouville that it is constant, hence trivial. It is also a sharp warning that global harmonic functions are scarce: to get interesting harmonic functions you generally need a boundary or a singularity somewhere.
There is no bounded, non-constant temperature distribution that is in equilibrium everywhere across all of empty space. Any candidate, no matter how cleverly shaped, must by Liouville be uniform. By contrast x^2 - y^2 is harmonic on the whole plane but unbounded, so it escapes the theorem — exactly as the theorem permits.
Bounded plus harmonic on all of space equals constant — global rigidity at its starkest.
Both hypotheses are essential: defined on ALL of space (a harmonic function on a bounded region can be bounded and wildly non-constant), and bounded (drop it and you get every polynomial like x^2 - y^2). It is a statement about entire harmonic functions, not local ones.