 Solution

Integration by parts gives the minimal surface equation for a graph in nondivergence form: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 $C_{\mathrm{loc}}^{2,\beta }$ 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 $\mathcal{L}{u}_k=0$, withFor $v=|Du{|}^2$ this yieldsIf $u\in C^2(\overline{\Omega })$, 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: