Geodesic Graphs with Constant Mean Curvature in Spheres

We study the existence and unicity of graphs with constant mean curvature in the Euclidean sphere of radius a. Given a compact domain , with some conditions, contained in a totally geodesic sphere S of and a real differentiable function on , we define the graph of in considering the height (x) on the minimizing geodesic joining the point x of to a fixed pole of . For a real number H such that |H| is bounded for a constant depending on the mean curvature of the boundary of and on a fixed number in (0,1), we prove that there exists a unique graph with constant mean curvature H and with boundary , whenever the diameter of is smaller than a constant depending on . If we have conditions on , that is, let be a graph over of a function, if |H| is bounded for a constant depending only on the mean curvature of and if the diameter of is smaller than a constant depending on H and , then we prove that there exists a unique graphs with mean curvature H and boundary . The existence of such a graphs is equivalent to assure the existence of the solution of a Dirichlet problem envolving a nonlinear elliptic operator.