On the existence of foliations by solutions to the exterior Dirichlet problem for the minimal surface equation

Given an exterior domain $\mathrm{\Omega }$ with $C^{2,\alpha }$ boundary in ${\mathbb{R}}^n$, $n\ge 3$, we obtain a $1$-parameter family $u_{\gamma}\in C^{\infty}\left(\Omega\right) $,$\left\vert \gamma\right\vert \leq\pi/2$, of solutions of the minimal surface equation such that, if $\left\vert \gamma\right\vert <\pi/2$,$u_{\gamma}\in C^{\infty}\left( \Omega\right) \cap C^{2,\alpha}\left( \overline{\Omega}\right) $,$u_{\gamma}|{\partial\Omega}=0$with$\max{\partial\Omega}\left\Vert \nabla u_{\gamma}\right\Vert =\tan\gamma$and,if$\left\vert \gamma\right\vert =\pi/2$,thegraphof$u_{\gamma}$ is contained in a $C^{1,1}$ manifold $M_{\gamma }\subset \bar{\mathrm{\Omega }}\times \mathbb{R}$ with $\partial M_{\gamma}=\partial\Omega$. Each of these functions is bounded and asymptotic to a constant $$c_{\gamma }=\underset{\parallel x\parallel \to \mathrm{\infty }}{\lim }{u}_{\gamma }\left(x\right)\mathrm{.}$$ The mappings $\gamma \to u_{\gamma }\left(x\right)$ (for fixed $x\in \mathrm{\Omega }$) and $\gamma \to c_{\gamma }$ are strictly increasing and bounded. The graphs of these functions foliate the open subset of ${\mathbb{R}}^{n+1}$ $$\{(x,z)\in \mathrm{\Omega }\times \mathbb{R}\text{, }-u_{\pi \mathrm{/}2}\left(x\right)<z<u_{\pi \mathrm{/}2}\left(x\right)\}\mathrm{.}$$ Moreover, if ${\mathbb{R}}^n\mathrm{\backslash }\mathrm{\Omega }$ satisfies the interior sphere condition of maximal radius $\rho$ and if $\mathrm{\partial }\mathrm{\Omega }$ is contained in a ball of minimal radius $\varrho$, then $$[0,{\sigma }_n\rho ]\subset [0,c_{\pi \mathrm{/}2}]\subset [0,{\sigma }_n\varrho ],$$ where $${\sigma }_n={\int }_1^{\mathrm{\infty }}\frac{dt}{\sqrt{t^{2(n-1)}-1}}\mathrm{.}$$ One of the above inclusions is an equality if and only if $\rho =\varrho$, $\mathrm{\Omega }$ is the exterior of a ball of radius $\rho$ and the solutions are radial.