arXiv subjects

Changyu Ren

Publications and source records attributed to Changyu Ren.

22 records · Page 2Linked to original sources

arXiv2017

The curvature estimates for convex solutions of some fully nonlinear Hessian type equations

The curvature estimates of quotient curvature equation do not always exist even for convex setting \cite{GRW}. Thus it is natural question to find other type of elliptic equations possessing curvature estimates. In this paper, we discuss the existence of curvature estimate for fully nonlinear elliptic equations defined by symmetric polynomials, mainlly, the linear combination of elementary symmetric polynomials.

math.AP↗

arXiv2016

On the curvature estimates for Hessian equations

The curvature estimates of $k$ curvature equations for general right hand side is a longstanding problem. In this paper, we totally solve the $n-1$ case and we also discuss some applications for our estimate.

math.AP↗

arXiv2015

An interior estimate for convex solutions and a rigidity theorem

We establish an interior $C^2$ estimate for $k+1$ convex solutions to Dirichlet problems of $k$-Hessian equations. We also use such estimate to obtain a rigidity theorem for $k+1$ convex entire solutions of $k$-Hessian equations in Euclidean space.

math.AP↗

arXiv2013

Global $C^2$ estimates for convex solutions of curvature equations

We establish $C^2$ a priori estimate for convex hypersurfaces whose principal curvatures $κ=(κ_1,..., κ_n)$ satisfying Weingarten curvature equation $σ_k(κ(X))=f(X,ν(X))$. We also obtain such estimate for admissible 2-convex hypersurfaces in the case $k=2$. Our estimates resolve a longstanding problem in geometric fully nonlinear elliptic equations.

math.AP↗