Boundary regularity for solutions of the equation of prescribed Gauss curvature

Abstract

We study the boundary regularity of convex solutions of the equation of prescribed Gauss curvature in a domain $\Omega \subset R^n$ in the case that the gradient of the solution is infinite on some relatively open, uniformly convex portion $\Gamma$ of $\partial \Omega$. Under suitable conditions on the data we show that near $\Gamma \times R$ the graph of $u$ is a smooth hypersurface (as a submanifold of $R^{n+1}$) and that $u{|}_{\Gamma }$ is smooth. In particular, $u$ is Hölder continuous with exponent $1/2$ near $Г$.