Liouville's theorem for harmonic functions

Remarks

Proof idea for a bounded function, $\abs{u} \le M$: by the gradient estimate on the ball $B(x, r)$, $\abs{\nabla u(x)} \le n M / r$ for every $r > 0$. Letting $r \to \infty$ gives $\nabla u = 0$ everywhere, so $u$ is constant. If $u$ is only bounded below, say $u \ge m$, apply Harnack's inequality to the nonnegative harmonic function $u - m$ on larger and larger balls. It shows that $u - m$ is bounded, and the first case applies. The theorem fails on proper subsets: on the half-space $\set{x_n > 0}$, $u = x_n$ is bounded below and not constant. Some one-sided condition is needed: $u(x, y) = x$ is harmonic on $\R^2$ and not constant. A harmonic function on $\R^n$ with $\abs{u(x)} \le C(1 + \abs{x})^k$ is a harmonic polynomial of degree at most $k$.

Going deeper

For a student

For $n = 2$ this is the real counterpart of Liouville's theorem for entire functions, and the two imply each other through harmonic conjugates. The theorem gives uniqueness of solutions of Poisson's equation on $\R^n$ that tend to $0$ at infinity. It is the prototype of Liouville theorems for elliptic equations and for harmonic maps. On $\R^n$ minus a point, or on other spaces, the analogous question leads to the distinction between parabolic and hyperbolic spaces.