On the Dirichlet problem for minimal graphs in hyperbolic space (Q919514)

The Dirichlet problem \(\Delta f-(f_ if_ j/(1+| df|^ 2))f_{ij}+n/f=0\) in \(\Omega\), \(f>0\) in \(\Omega\), \(f=0\) on \(\partial \Omega\) is investigated. As a main result the following theorem is proved: If \(\partial \Omega\) is of class \(C^{k,\alpha}\), then the graph (f) is a \(C^{k,\alpha}\) hypersurface with boundary for either (i) \(1\leq k\leq n-1\) and \(0\leq \alpha \leq 1\) or (ii) \(n\leq k\leq \infty\) and \(0<\alpha <1.\) To show the higher regularity, the author studies carefully a degenerate elliptic partial differential equation \[ y\{\Delta u-(u_ iu_ j/(1+| du|^ 2))u_{ij}\}-nu_ j=0\text{ in } y>0. \]