An extremal case of the equation of prescribed Weingarten curvature

Abstract

In this paper we investigate the existence and regularity of solutions to a Dirichlet problem for a Hessian quotient equation on the sphere. The equation in question arises as the determining equation for the support function of a convex surface which is required to meet a given enclosing cylinder tangentially and whose k-th Weingarten curvature is a given function. This is a generalization of a Gaussian curvature problem treated in [13]. Essentially given ${\Omega \subset \mathbb{R}^n}$ we seek a convex function u such that graph(u) has a prescribed k-th curvature ψ and |Du(x)| → ∞ as x → ∂Ω. Under certain regularity assumptions on ψ and Ω we are able to demonstrate the existence of a solution whose graph is C ^3,α provided that ${\psi^{-\frac{1}{k}} = \psi^{-\frac{1}{k}}(x, \nu)}$ is convex in x and a certain compatibility condition between ψ| _∂Ω and Ω is satisfied.