On the existence of foliations by solutions to the exterior Dirichlet problem for the minimal surface equation
Given an exterior domain $Ω$ with $C^{2,α}$ boundary in $\mathbb{R}^{n}$, $n\geq3$, we obtain a $1$-parameter family $u_γ\in C^{\infty}\left(Ω\right) $, $\left\vert γ\right\vert \leqπ/2$, of solutions of the minimal surface equation such that, if $\left\vert γ\right\vert <π/2$, $u_γ\in C^{\infty}\left( Ω\right) \cap C^{2,α}\left( \overlineΩ\right) $, $u_γ|_{\partialΩ}=0$ with $\max_{\partialΩ}\left\Vert \nabla u_γ\right\Vert =\tanγ$ and, if $\left\vert γ\right\vert =π/2$, the graph of $u_γ$ is contained in a $C^{1,1}$ manifold $M_γ\subset\overlineΩ\times\mathbb{R}$ with $\partial M_γ=\partialΩ$. Each of these functions is bounded and asymptotic to a constant \[ c_γ=\lim_{\left\Vert x\right\Vert \rightarrow\infty}u_γ\left( x\right) . \] The mappings $γ\rightarrow u_γ\left( x\right) $ (for fixed $x\inΩ$) and $γ\rightarrow c_γ$ are strictly increasing and bounded. The graphs of these functions foliate the open subset of $\mathbb{R}^{n+1}$ \[ \left\{ \left( x,z\right) \inΩ\times\mathbb{R}\text{, }-u_{π/2}\left( x\right) <z<u_{π/2}\left( x\right) \right\} . \] Moreover, if $\mathbb{R}^{n}\backslashΩ$ satisfies the interior sphere condition of maximal radius $ρ$ and if $\partialΩ$ is contained in a ball of minimal radius $\varrho$, then \[ \left[ 0,σ_{n}ρ\right] \subset\left[ 0,c_{π/2}\right] \subset\left[ 0,σ_{n}\varrho\right] , \] where \[ σ_{n}=\int_{1}^{\infty}\frac{dt}{\sqrt{t^{2\left( n-1\right) }-1}}. \] One of the above inclusions is an equality if and only if $ρ=\varrho$, $Ω$ is the exterior of a ball of radius $ρ$ and the solutions are radial.