Let $\Omega \subset \mathbf{R}^n$ be a smooth, bounded domain. The Dirichlet problem for the minimal surface equation
\begin{equation}
(1 + \lvert Du \rvert^2) \Delta u - D_i u D_j u D_{ij} u = 0
\end{equation}
is (uniquely) solvable provided the boundary data $g \in C^{1,\alpha}(\partial \Omega)$ is small enough, say
\begin{equation}
\lvert g \rvert_{C^{1,\alpha}} < \delta.
\end{equation}

> What estimates are available for the Dirichlet-to-Neumann operator? (If needed, you may impose $g$ smaller yet than the above.)

For example, is it true, provided $\lvert g \rvert_{C^{1,\alpha}} < \delta'$ say, that
\begin{equation}
\lvert \partial u/ \partial \nu \rvert_{C^{0,\alpha}}
\leq C \lvert g \rvert_{C^{1,\alpha}}?
\end{equation}
- The catenoid&mdash;or really half of it&mdash;considered on an annular domain demonstrates that some smallness is required.
- It seems like it might be a (direct?) consequence of Allard boundary regularity theorems, but I am not sure... It could be easier as well!