J. Ripoll, Some characterization, uniqueness and existence results for Euclidean graphs of constant mean curvature with planar boundary, PAC J MATH, 198(1), 2001, pp. 175-196
We establish the existence and uniqueness of solutions to the Dirichlet pro
blem for the cmc surface equation, including the minimal one, for zero boun
dary data, in certain domains of the plane. We obtain results that characte
rize the sphere and cmc graphs among compact embedded cmc surfaces with pla
nar boundary satisfying certain geometric conditions. We also find conditio
ns that imply that a compact embedded cmc surface which is a graph near the
boundary is indeed a global graph.