I have a number of results on the existence of solutions to the Plateau problem for surfaces of prescribed mean curvature. One of the more interesting papers  holds for surfaces in higher codimension, and introduces the natural concept of a surface with prescribed mean curvature vector. Given an alternating (2,1)-tensor H on a Riemannian manifold (such as Rn), a surface is said to have prescribed mean curvature vector H if at each point x, the mean curvature vector of the surface equals the value of H on an orthonormal basis of the tangent plane to the surface. Choose a point p in the manifold, and let b2 be an upper bound for sectional curvatures. One result is that if H is formed by raising an index on an exact 3-form and has radial component at most b cot(br) when evaluated on any orthonormal pair, where r is the distance to p, then any Jordan curve in the ball of radius r centered at p is the boundary of a branched disk of prescribed mean curvature vector H. These bounds are sharp . Analogous theorems for currents of dimension greater than 2 have been proved by Klaus Steffen and Frank Duzaar.
 Existence of Surfaces with Prescribed Mean Curvature Vector,
Math. Z. 131, 117-140 (1973).
 Necessary Conditions for Submanifolds and Currents with Prescribed Mean Curvature Vector, Annals of Mathematics Studies 103, 225-242 (1983).