The Neumann problem of Hessian quotient equations

Abstract

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.

The Neumann problem for a class of fully nonlinear elliptic partial differential equations

“…One can find a generalized inequality and the proof in [6]. For completeness we give a proof for our case as same as in [22].…”

“…Recently, Ma and Qiu [22] gave a positive answer to this problem and solved the the Neumann problem of k-Hessian equations in uniformly convex domains. After their work, the research on the Neumann problem of other equatios has made many progresses(see [23] [6] [2] [33]).…”

The Neumann problem for a type of fully nonlinear complex equations

“…Until recently, following the breakthrough work of Ma-Qiu [30], who solved the Neumann problem for k-Hessian equations on uniformly convex domains in R n , some literatures have appeared, such as Chen-Wei [6] for the complex Hessian quotient equations and Chen-Chen-Mei-Xiang [3] for a class of mixed complex Hessian equations. See [7] and [2] for the corresponding problems in the real situations. Comparing to the complex fully nonlinear equations, the works about the real fully nonlinear equations are more abundant.…”

The Neumann Problem of Complex Special Lagrangian Equations with Supercritical Phase

“…In 1986, Lions-Trudinger-Urbas solved the Neumann problem of Monge-Ampère equation in the celebrated paper [23]. Recently, Ma-Qiu [24] solved the the Neumann problem of k-Hessian equations, and Chen-Zhang [6] generalized the result to the the Neumann problem of Hessian quotient equations. For the Neumann problem of special Lagrangian equations with supercritical phase, Chen-Ma-Wei [5] got the existence theorem.…”