Constant Mean Curvature Graphs on Unbounded Convex Domains

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 . We also establish an existence result for convex bounded domains contained in a strip.