Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 5 / 5 / c / Solution

The local Harnack inequality for harmonic functions states that if $u\ge 0$ is harmonic on a neighbourhood of $\overline{B_{4r}(a)}$, thenMore generally, for any compact $K$ in a connected open domain $\Omega$, there is $C=C(K,\Omega ,n)$ such that ${\sup }_{K}u\le C{\inf }_{K}u$ for every nonnegative harmonic $u$ on $\Omega$. Positivity is essential; the assertion is not for arbitrary sign-changing functions.

First prove the mean value property for harmonic functions. For a smooth harmonic function the spherical mean has derivativeso it equals $u(x)$; integration in the radius gives the ball mean property. The preceding Weyl lemma supplies smoothness if harmonicity was initially distributional.

For $x,z\in B_r(a)$ the inclusion $B_r(x)\subset B_{3r}(z)\subset B_{4r}(a)$ and nonnegativity giveTaking the supremum over $x$ and the infimum over $z$ proves the local estimate directly. To obtain the compact-set form, connect centres of a finite ball cover of $K$ to a fixed interior point by paths in $\Omega$. The compact union of those paths stays a positive distance from the complement of $\Omega$. Subdivide the paths into finitely many steps small enough that the local estimate applies at consecutive points. Multiply the finitely many comparison constants, first from one point of $K$ to the reference point and then to any other point of $K$. This gives a constant independent of $u$. In particular a nonnegative harmonic function is either everywhere positive or identically zero on each connected component.