Global continuity estimates for two dimensional graphs of prescribed Gauss curvature

We prove the global Hölder continuity of convex solutions u ∈ C³ (Ω) of the equation of prescribed positive Gauss curvature in a bounded convex domain Ω ⊂ R² with ∂Ω ∈ C^1,β for some β ∈ (0, 1], We also obtain better regularity for the trace of u on ∂Ω. In the special case β= 1 we show that u ∈ C^{0, 1/2} (Ω ̄)and u|C ^{0, 2/3}(∂Ω). We also investigate the global continuity of solutions in C¹ domains and construct an example showing that global continuity need not hold in general convex domains.