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

Given an exterior domain $\Omega$ with $C^{2,\alpha}$ boundary in $\mathbb{R}^{n}$, $n\geq3$, 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 $\left\vert \gamma\right\vert =\pi/2$, the graph of $u_{\gamma}$ is contained in a $C^{1,1}$ manifold $M_{\gamma}\subset\overline{\Omega}\times\mathbb{R}$ with $\partial M_{\gamma}=\partial\Omega$. Each of these functions is bounded and asymptotic to a constant $$c_{\gamma}=\lim_{\left\Vert x\right\Vert \rightarrow\infty}u_{\gamma}\left( x\right) .$$ The mappings $\gamma\rightarrow u_{\gamma}\left( x\right)$ (for fixed $x\in\Omega$) and $\gamma\rightarrow c_{\gamma}$ are strictly increasing and bounded. The graphs of these functions foliate the open subset of $\mathbb{R}^{n+1}$ $$\left\{ \left( x,z\right) \in\Omega\times\mathbb{R}\text{, }-u_{\pi /2}\left( x\right) <z<u_{\pi/2}\left( x\right) \right\} .$$ Moreover, if $\mathbb{R}^{n}\backslash\Omega$ satisfies the interior sphere condition of maximal radius $\rho$ and if $\partial\Omega$ is contained in a ball of minimal radius $\varrho$, then $$\left[ 0,\sigma_{n}\rho\right] \subset\left[ 0,c_{\pi/2}\right] \subset\left[ 0,\sigma_{n}\varrho\right] ,$$ where $$\sigma_{n}=\int_{1}^{\infty}\frac{dt}{\sqrt{t^{2\left( n-1\right) }-1}}.$$ One of the above inclusions is an equality if and only if $\rho=\varrho$, $\Omega$ is the exterior of a ball of radius $\rho$ and the solutions are radial.