## 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 Ω ⊂ ℝ^{n} we seek a convex function u such that graph(u) has a prescribed k-th curvature ψ and {pipe}Du(x){pipe} → ∞ 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, is convex in x and a certain compatibility condition between ψ{pipe}_{∂Ω} and Ω is satisfied.

Original language | English |
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Pages (from-to) | 277-291 |

Number of pages | 15 |

Journal | Calculus of Variations and Partial Differential Equations |

Volume | 48 |

Issue number | 3-4 |

DOIs | |

Publication status | Published - Nov 2013 |