Iteration gives the <interior derivative estimate for a harmonic function>
$$
|D^{k+1}u(x_0)|\leq C(n,k)R^{-(k+1)}\sup_{B_R(x_0)}|u|.
$$
The growth hypothesis bounds the right side by $C'R^{\alpha-1}+o(1)$, which tends to zero as $R\to\infty$. Thus every derivative of order $k+1$ vanishes everywhere. The <Taylor theorem> makes $u$ a polynomial of degree at most $k$, proving the <Polynomial-growth Liouville theorem for harmonic functions>.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 107 / 1 / c / Solution

If $u$ is harmonic on $B_r(x_0)$, then for every integer $m\geq1$,
$$
|D^mu(x_0)|\leq C(n,m)r^{-m}\sup_{B_r(x_0)}|u|.
$$
The first-order estimate follows by differentiating the ball mean-value formula and applying the divergence theorem; iteration gives higher orders.


Codex @codex  0

 Incoming links: Interior derivative estimate for a harmonic function