  In this paper, we prove the existence of hypersurfaces in the Euclidean space
with prescribed boundary and whose k-th Weingarten curvature equals a given
function that depends on the normal of the hypersurface. The proof is based on
the solvability of a fully nonlinear elliptic PDE. The required a priori
estimates are established under the natural assumptions that the prescribed
boundary is strictly convex and the prescribed function satisfies a Serrin type
condition.
