Inequality for non negative harmonic functions

Question

I want to prove that if we have a non negative harmonic function $u$ in a domain $\Omega\subset\mathbb{R}^N$, then for every subdomain $\Omega'\subset\Omega$ such as $\overline{\Omega'}\subset \Omega$, there exists some constant $C$ depending on $N,\Omega$ and $\Omega'$ such as $$\operatorname{sup}u_{\Omega'}\leq c\operatorname{inf}u_{\Omega'}.$$ I've been able to prove that if for some $z\in \Omega$ and $R>0$ we have that $ B(z,4R)\subset \Omega$, then for every $x_1,x_2\in B(z,R)$ we have that $$u(x_1)\leq 3^Nu(x_2),$$ which is a result i think it's useful in the prove. Anyway i don't know how to properly apply it in the proof of the result. Any help will be very appreciated.

Answer 1

Since $\Omega'$ is a domain, there exist some $R>0$ such as for every $z\in \Omega'$ we have that $B(z,4R)\subset \Omega'$. Then we have $$\overline{\Omega'}\subset \bigcup_{z\in \overline{\Omega'}}B(z,4R).$$ Since $\overline{\Omega'}$ is compact we can find a finite subcover of it, precisely we can find $z_1,\ldots,z_m\in \overline{\Omega'}$ such as $$\overline{\Omega'}\subset\bigcup_{i=1}^mB(z_i,4R).$$ Then, given $x,x'\in\overline{\Omega'},$ we can find a chain of $j$ balls we can denote by $B_i$ connecting both points. Now choosing $x_i$ in the intersection of the consecutive ball $B_i$ and $B_{i+1}$, we can apply $j$ times the result you have proven to get $$u(x)\leq 3^Nu(x_1)\leq\cdots\leq 3^{jN}u(x').$$ Now taking $C=3^{jN}$( which depends on $N,\Omega,\Omega'$ as desired), and $x=\operatorname{sup}_{\Omega'} u, x'=\operatorname{inf}_{\Omega'}u,$ we have the desired result.