Global continuity estimates for two dimensional graphs of prescribed Gauss curvature

Abstract

We prove the global Holder continuity of convex solutions u ∈ C3(Ω) of the equation of prescribed positive Gauss curvature in a bounded convex domain 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 and u|∂Ω ∈ C0,2/3(∂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.