I'm taking my first theoretical PDE course in a year and am bashing my head against a rock with this problem.
Prove that there exists a constant $C$, depending only on $n$, such that
$\max_{B(0,1)}|u| ≤ C \max_{∂B(0,1)}|g| + \max_{B(0,1)}| f |)$
whenever $u$ is a smooth solution of
$-\Delta u = f$ in $B(0,1)$
$u = g$ on $∂B(0,1)$
I found similar problems online but their proofs do not make any sense. Does anyone have any suggestions?