Harnack inequality on compact subsets

If $u$ has an interior zero, it vanishes throughout $\mathrm{\Omega }$, and the inequality is immediate. Otherwise it is strictly positive. Fix the finite admissible-ball family of the chain lemma, of size $N$, independently of $u$.

If $x,y\in B_r(a)$ from this family, then $B_{3r}(y)\subset B_{4r}(a)\subset \mathrm{\Omega }$ and $\mathrm{\mid }x-y\mathrm{\mid }<2r$. Ball Harnack gives $u(x)\le 3^{n}u(y)$. Thus any two points of one of the small balls are comparable by the same constant.

For $x,y\in K$, connect balls containing them by a simple path in the finite connected graph, with at most $N$ vertices. Choose one point in each of its finitely many overlaps and apply the local comparison in successive balls. It gives $u(x)\le 3^{nN}u(y)$. Taking the supremum in $x$ and the infimum in $y$ proves the claim with $C=3^{nN}$.