Surfaces of prescribed Weingarten curvature tangential to a cone

摘要

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 arises as the determining equation for the support function of a convex surface which is required to meet a given enclosing cone tangentially and whose kthWeingarten curvature is a prescribed function ψ. This is a generalization of a related problem treated in [7] and is motivated by results from the theory of curvature flows [16, 17]. In the general case, we are able to obtain C1 estimates provided ψ satisfies a certain weak asymptotic growth condition. Under further regularity assumptions we are able to demonstrate, via a priori estimates and the continuity method, the existence of bounded C~(2,α) solutions under a convexity condition on ψ. We also demonstrate conditions under which no solution can exist.