Suppose $\Omega$ is an open, bounded, connected set. Let $f$ be a continuous function on $\overline\Omega$ and $\mathcal{F}$ be the family of harmonic functions on $\Omega$ that belong to $C(\overline\Omega)$ such that $u \leq f$ on the boundary $\partial\Omega$.
Can we show that $\mathcal{F}$ is a normal family?
Each $u$ should attain a minimum and maximum and it seems like one could apply the Arzela-Ascoli Theorem here, but how could we prove uniform boundedness and equicontinuity of the family?