The boundary value Minkowski problem for Weingarten curvatures

Abstract

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.