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 $\mathbb {R}^{n}$, $n\geq 3$, 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 $\left \vert \gamma \right \vert =\pi /2$, 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.

These foliations were studied by E. Kuwert in [Ann. Inst. H. Poincaré Anal. Non Linéaire 10 (1993), pp. 445–451] and our result answers a natural question about the existence of such foliations which was not touched by Kuwert.