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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 \(\mathbb {R}^{n}\) , \(n\geq 2\) . 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 \(\sup _{\partial \Omega }\left \vert \operatorname {grad} u_{s}\right \vert =\left \vert s\right \vert \) . Additionally, we provide further information on the behavior of these solutions at infinity.

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The Dirichlet Problem for the Minimal Surface Equation on Unbounded Helicoidal Domains of \( \mathbb {R}^{m}\)

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  • 发表日期：

    2025年

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  • Ari Aiolfi,
  • Caroline Assmann,
  • Jaime Ripoll

摘要

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 \(\mathbb {R}^{n}\) , \(n\geq 2\) . 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 \(\sup _{\partial \Omega }\left \vert \operatorname {grad} u_{s}\right \vert =\left \vert s\right \vert \) . Additionally, we provide further information on the behavior of these solutions at infinity.