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

Abstract

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