The boundary value Minkowski problem for Weingarten curvatures

TL;DR

This paper proves the existence of hypersurfaces with prescribed boundary and Weingarten curvature in Euclidean space by solving a nonlinear elliptic PDE, under convexity and Serrin-type conditions.

Contribution

It establishes the existence of hypersurfaces with prescribed boundary and Weingarten curvature using new a priori estimates and PDE techniques.

Findings

Existence of hypersurfaces with prescribed boundary and Weingarten curvature.

Development of a priori estimates under convexity and Serrin conditions.

Solution of a fully nonlinear elliptic PDE for the problem.

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.