Constant Mean Curvature Graphs on Unbounded Convex Domains

Abstract

In this paper, we consider the Dirichlet problem for the constant mean curvature equation on an unbounded convex planar domain Ω. Let H>0. We prove that there exists a graph with constant mean curvature H and with boundary ∂Ω if and only if Ω is included in an infinite strip of width 1H. We also establish an existence result for convex bounded domains contained in a strip.