Pointwise estimate for solutions of the laplace equation on bounded domains
Let $\Omega$ be a bounded open subset of $\mathbb{R}^ n$ . Prove that there exists a constant $C$, depending only on $\Omega$, such that
$$\max |u| \leq C (\max_{\partial \Omega }|g|+\max_{\overline{\Omega} }|f|), $$
where $u$ is a solutions of the PDE
$$ -\Delta u=f $$ with $u=g$ on $\partial \Omega$.
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Erdos
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