= Gradient maximum principle for a minimal graph
For a $C^2$ solution $u$ of the <minimal surface equation for a graph> on the closure of a bounded domain,
$$
\sup_\Omega|Du|=\sup_{\partial\Omega}|Du|.
$$
Write $a^{ij}(Du)u_{ij}=0$ with $a^{ij}(p)=\delta_{ij}-p_ip_j/(1+|p|^2)$. Its differentiated operator $\mathcal L=a^{ij}D_{ij}+b^\ell D_\ell$, where $b^\ell=(\partial a^{ij}/\partial p_\ell)u_{ij}$, satisfies
$$
\mathcal L|Du|^2=2\sum_k a^{ij}u_{ki}u_{kj}\geq0.
$$
The <weak maximum principle for elliptic operators> gives the result. Interior <Schauder estimates> justify differentiating a solution initially assumed only $C^2$.
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