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 $\mathbb{R}^{n}$, $n\geq2$. 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\vert \operatorname{grad}u_{s}\right\vert =\left\vert s\right\vert $. Additionally, we provide further information on the behavior of these solutions at infinity.