The Neumann Problem of Complex Hessian Quotient Equations

Abstract

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.

A class of fully nonlinear equations

“…Proof. In the paper [2], inequality (2.12) has been proved. Here we omit the details of the proof and just prove the (2.13) based on (2.12).…”

“…In this section, we introduce some notations and key lemmas which will be used later, and omit some details of the proof for lemmas. We refer the readers to see the details in [2,6].…”

“…The following lemmas play an important role in the proof of derivative estimates. The idea of the proof for these lemmas comes from the paper in [2]. Lemma 2.5.…”

“…Here we omit the details of the proof and just prove the (2.13) based on (2.12). We know from Lemma 2.6 in [2],…”

“…Recently, Ma-Qiu [13] have proved the existence of a classical solution to a Neumann boundary problem for Hessian equations in uniformly convex domain. Chen-Zhang [2] obtain some important inequalities of Hessian quotient operators, and establish the existence theorem. Jiang-Trudinger [7,8,9], studied the general oblique boundary problem for augmented Hessian equations with some regular conditions and concavity conditions.…”