Surfaces of prescribed Weingarten curvature tangential to a cone

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 $k$th Weingarten curvature is a prescribed function $\psi$. 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 $C^1$ estimates provided $\psi$ 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 $C^{2,\alpha}$ solutions under a convexity condition on $\psi$. We also demonstrate conditions under which no solution can exist.

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Published 22 September 2014