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In this paper, we obtain some important inequalities of Hessian quotient operators, and global [Formula: see text] estimates of the Neumann problem of Hessian quotient equations. By the method of continuity, we establish the existence theorem of [Formula: see text]-admissible solutions of the Neumann problem of Hessian quotient equations.

In this paper, we consider the Neumann problem of a class of mixed complex Hessian equations σ k (∂∂u) = k−1 l=0 α l (x)σ l (∂∂u) with α l positive and 2 ≤ k ≤ n, and establish the global C 1 estimates and reduce the global second derivative estimate to the estimate of double normal second derivatives on the boundary. In particular, we can prove the global C 2 estimates and the existence theorems when k = n.Mathematical Subject Classification (2010): Primary 35J60, Secondary 35B45.

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Abstract. In this paper, we introduce a new auxiliary function, and establish the interior C 2 estimate for Monge-Ampère equation in dimension n = 2, which was firstly proved by Heinz [5] by a geometric method.

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In this paper, we introduce a new auxiliary function, and establish the interior C 2 estimate for prescribed Gauss curvature equation in dimension two.

For the Monge-Ampère equation det D 2 u = 1, we find new auxiliary curvature functions which attain respective maximum on the boundary. Moreover, we obtain the upper bounded estimates for the Gauss curvature and mean curvature of the level sets for the solution to this equation. 2000 Mathematics Subject Classification: 35J65.