Gradient estimates for harmonic functions

Remarks

Proof idea for part 1: each partial derivative $\partial_i u$ is harmonic, so by the mean value property over the ball and the divergence theorem, $$\partial_i u(x_0) = \frac{1}{\omega_n r^n} \int_{B(x_0, r)} \partial_i u \, dy = \frac{1}{\omega_n r^n} \int_{\partial B(x_0, r)} u \, \nu_i \, dS.$$ The last integral is at most $\max \abs{u}$ times the area $n \omega_n r^{n-1}$ of the sphere, which gives $\abs{\partial_i u(x_0)} \le \frac{n}{r} \max \abs{u}$. The same argument for the derivative in the direction of $\nabla u(x_0)$ gives the bound for $\abs{\nabla u(x_0)}$. Part 2 follows by induction on $k$, applying the same argument on balls of radius $r/k$. Example: $u(x, y) = x$ on the unit disc has $\abs{\nabla u} = 1$ and $\max \abs{u} = 1$, within the bound $2$ of part 1. The estimates are interior: they control derivatives at the centre by the size of $u$ on the ball, and they fail near the boundary of the domain. For instance $u_m(x, y) = \operatorname{Re}(x + iy)^m$ on the unit disc has $\abs{u_m} \le 1$ but $\abs{\nabla u_m(1, 0)} = m$.

Going deeper

For a student

These estimates give Liouville's theorem, the real analyticity of harmonic functions, and the compactness of bounded families of harmonic functions through the Arzelà–Ascoli theorem. They are the model for interior Schauder and Sobolev estimates for general elliptic equations. For nonnegative harmonic functions there is a sharper form, $\abs{\nabla u(x_0)} \le \frac{n}{r} u(x_0)$, which leads to Harnack's inequality.