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Given an exterior domain $\Omega$ with $C^{2,\alpha}$ boundary in$\mathbb{R}^{n}$, $n\geq3$, we obtain a $1$-parameter family $u_{\gamma}\inC^{\infty}\left(\Omega\right) $, $\left\vert \gamma\right\vert \leq\pi/2$, ofsolutions of the minimal surface equation such that, if $\left\vert\gamma\right\vert <\pi/2$, $u_{\gamma}\in C^{\infty}\left( \Omega\right) \capC^{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 $u_{\gamma}$ iscontained in a $C^{1,1}$ manifold$M_{\gamma}\subset\overline{\Omega}\times\mathbb{R}$ with $\partialM_{\gamma}=\partial\Omega$. Each of these functions is bounded and asymptoticto 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 graphsof 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 ofmaximal radius $\rho$ and if $\partial\Omega$ is contained in a ball of minimalradius $\varrho$, then \[ \left[ 0,\sigma_{n}\rho\right] \subset\left[0,c_{\pi/2}\right] \subset\left[ 0,\sigma_{n}\varrho\right] , \] where \[\sigma_{n}=\int_{1}^{\infty}\frac{dt}{\sqrt{t^{2\left( n-1\right) }-1}}. \] Oneof 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.

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