= Solution
Integration by parts gives the <minimal surface equation for a graph> in nondivergence form:
$$
a^{ij}(Du)u_{ij}=0,\qquad
a^{ij}(p)=\delta_{ij}-\frac{p_ip_j}{1+|p|^2}.
$$
Its eigenvalues are between $(1+|p|^2)^{-1}$ and one. On each relatively compact subdomain, $Du$ is bounded, so it is a <uniformly elliptic operator>. Since $u\in C^2$, its coefficient functions are locally Lipschitz. The <interior Schauder estimate> gives $C^{2,\beta}_{\rm loc}$ regularity for every $0<\beta<1$; differentiating and repeatedly applying regularity estimates gives $\boxed{u\in C^\infty(\Omega)}$.
Differentiate in direction $k$. Each $u_k$ satisfies $\mathcal Lu_k=0$, with
$$
\mathcal L=a^{ij}D_{ij}+b^\ell D_\ell,\qquad
b^\ell=\frac{\partial a^{ij}}{\partial p_\ell}(Du)u_{ij}.
$$
For $v=|Du|^2$ this yields
$$
\mathcal Lv=2\sum_k a^{ij}u_{ki}u_{kj}\geq0.
$$
If $u\in C^2(\overline\Omega)$, the principal coefficients are uniformly elliptic globally and the drift is bounded. The <weak maximum principle for elliptic operators> bounds $v$ by its boundary maximum. Boundary values are limits of interior values by <continuity>, proving the <gradient maximum principle for a minimal graph>:
$$
\boxed{\sup_\Omega|Du|=\sup_{\partial\Omega}|Du|.}
$$
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