Boundary behavior of a nonparametric surface of prescribed mean curvature near a reentrant corner

Abstract:

Let $\Omega$ be an open set in ${{\mathbf {R}}^2}$ which is locally convex at each point of its boundary except one, say $(0,0)$. Under certain mild assumptions, the solution of a prescribed mean curvature equation on $\Omega$ behaves as follows: All radial limits of the solution from directions in $\Omega$ exist at $(0,0)$, these limits are not identical, and the limits from a certain half-space $(H)$ are identical. In particular, the restriction of the solution to $\Omega \cap H$ is the solution of an appropriate Dirichlet problem.