The Dirichlet problem for the minimal surface equation on unbounded helicoidal domains of ${\mathbb{R}}^m$

Abstract

We consider a helicoidal group $G$ in ${\mathbb{R}}^{n+1}$ and unbounded $G$-invariant $C^{2,\alpha }$-domains $\Omega \subset {\mathbb{R}}^{n+1}$ whose helicoidal projections are exterior domains in ${\mathbb{R}}^n$, $n\ge 2$. We show that for all $s\in \mathbb{R}$, there exists a $G$-invariant solution $u_s\in C^{2,\alpha }\left(\overline{\Omega }\right)$ of the Dirichlet problem for the minimal surface equation with zero boundary data which satisfies ${\sup }_{\partial \Omega }\left|\mathrm{grad}{u}_s\right|=\left|s\right|$. Additionally, we provide further information on the behavior of these solutions at infinity.