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—or really half of it—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 too; maybe it's just a Schauder estimate?
