The Neumann Problem of Complex Hessian Quotient Equations

Abstract

Abstract In this paper, we consider the Neumann problem of complex Hessian quotient equations $\frac{\sigma _k (\partial \bar{\partial } u)}{\sigma _l (\partial \bar{\partial } u)} = f(z)$ with $0 \leq l < k \leq n$ and establish the global $C^1$ estimates and reduce the global 2nd derivative estimate to the estimate of double normal 2nd derivatives on the boundary. In particular, we can prove the global $C^2$ estimates and the existence theorem for the Neumann problem of complex Hessian quotient equations $\frac{\sigma _n (\partial \bar{\partial } u)}{\sigma _l (\partial \bar{\partial } u)} = f(z)$ with $0 \leq l < n$ by the method of continuity.