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

Abstract

Given an exterior domain $\Omega$ with $C^{2,\alpha }$ boundary in ${\mathbb{R}}^n$, $n\ge 3$, we obtain a $1$-parameter family $u_{\gamma }\in C^{\infty }\left(\Omega \right)$, $\left|\gamma \right|\le \pi /2$, of solutions of the minimal surface equation such that, if $\left|\gamma \right|<\pi /2$, $u_{\gamma }\in C^{\infty }\left(\Omega \right)\cap C^{2,\alpha }\left(\overline{\Omega }\right)$, $u_{\gamma }{|}_{\partial \Omega }=0$ with ${\max }_{\partial \Omega }\Vert \nabla u_{\gamma }\Vert =\tan \gamma$ and, if $\left|\gamma \right|=\pi /2$, 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 \[ c_γ=\lim_{\left\Vert x\right\Vert \rightarrow\infty}u_γ\left( x\right) . \] The mappings $\gamma \to u_{\gamma }\left(x\right)$ (for fixed $x\in \Omega$) and $\gamma \to c_{\gamma }$ are strictly increasing and bounded. The graphs of these functions foliate the open subset of ${\mathbb{R}}^{n+1}$ \[ \left\{ \left( x,z\right) \inΩ\times\mathbb{R}\text{, }-u_{π/2}\left( x\right) <z<u_{π/2}\left( x\right) \right\} . \] Moreover, if ${\mathbb{R}}^n\backslash \Omega$ satisfies the interior sphere condition of maximal radius $\rho$ and if $\partial \Omega$ is contained in a ball of minimal radius $\varrho$, then \[ \left[ 0,σ_{n}ρ\right] \subset\left[ 0,c_{π/2}\right] \subset\left[ 0,σ_{n}\varrho\right] , \] 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 $\rho =\varrho$, $\Omega$ is the exterior of a ball of radius $\rho$ and the solutions are radial.

To appear on Proceedings of the American Mathematical Society