Existence and uniqueness of minimal graphs in hyperbolic space (Q5929578)

Let \({\mathbb H}^{n+1}\) be (n+1)-dimensional hyperbolic space form, \(\partial_{\infty}{\mathbb H}^{n+1}={\mathbb R}^n\) in half space model of \({\mathbb H}^{n+1}\). The authors solve the Dirichlet problem of minimal surfaces equation in \({\mathbb H}^{n+1}\) over bounded domains \(\Omega \subset\partial_{\infty}{\mathbb H}^{n+1}={\mathbb R}^n\), namely the prove: NEWLINENEWLINENEWLINE1. when \(n=2\), \(\Omega\) is a compact \(C^0\) convex domain, then the minimal surface equation admits a unique solution for given continuous boundary value data;NEWLINENEWLINENEWLINE2. when \(n\geq 3\), \(\Omega \) is a bounded strictly convex domain with \(C^{2,\alpha}\) (\(\alpha\in (0,1)\)) boundary, then the minimal surface equation admits a unique solution for given \(C^{2,\alpha}\) boundary value data. NEWLINENEWLINENEWLINEWhen \(\Omega\) is unbounded, by using the Perron process the authors also obtain several existence and uniqueness theorems.