Second Order Derivative Estimates of the Solutions of a Class of Monge-Ampère Equations

In this paper, we consider a class of Monge-Amp`ere equations in relative differential geometry. Given these equations with zero boundary values in a smooth strictly convex bounded domain, we obtain second order derivative estimates of the convex solutions.     

Riemann流形上的抛物型Monge-Ampère方程

This paper deals with a class of parabolic Monge-Ampère equation on Riemannian manifolds. The existence and uniqueness of the solution to the first initial-boundary value problem for the equation are established.     

An Initial-Boundary Value Problem for Parabolic Monge-Ampère Equation in Mathematical Finance

This paper deals with some parabolic Monge-Ampère equation raised from mathematical finance: V_sV_(yy)+ryV_yV_(yy)-θV_y~2= 0(V_(yy) 0). The existence and uniqueness of smooth solution to its initial-boundary value problem with some requirement is obtained.     

Regularity Properties of the Degenerate Monge-Ampère Equations on Compact K(a)hler Manifolds

The author establishes a result concerning the regularity properties of the degenerate complex Monge-Ampère equations on compact K(a)hler manifolds.     

Curvature Estimates for the Level Sets of Solutions to the Monge-Ampère Equation det D~2u = 1

For the Monge-Amp`ere equation det D2u = 1, the authors find new auxiliary curvature functions which attain their respective maxima on the boundary. Moreover, the upper bounded estimates for the Gauss curvature and the mean curvature of the level sets for the solution to this equation are obtained.     

Regularity Properties of the Degenerate Monge-Ampère Equations on Compact Khler Manifolds

The author establishes a result concerning the regularity properties of the degenerate complex Monge-Ampère equations on compact Khler manifolds.     

一类Monge-Ampère方程的特征值问题

对一类Monge-Ampère方程的特征值问题进行了研究．通过移动平面法证明了在凸对称区域内,Dirichlet问题的C~2凹(凸)解一定是对称的．进而通过对常微分方程和椭圆形偏微分方程的讨论,得到一类n维单位球上特征值问题的非平凡解的存在性和正则性结果．     

关于复Monge-Ampère方程的一些研究

过去50年间,复Monge-Ampère方程是一个被广泛研究的主题,它在复分析、复几何和K(a)hler几何中有很多重要的应用.本文首先回顾一些经典结果,然后着重介绍关于复Monge-Ampère方程的Liouville定理、C1,α估计和C2,α估计,以及其在Sasakian几何中的一些应用.这些结果是与S.Dine...     

黎曼流形上具有Neumann边界条件的Monge-Ampère型方程

文章研究了黎曼流形上具有N eumann边界条件的Monge-Ampère型方程的全局正则性,并将其在欧几里得空间中的主要结论推广到了曲面空间.     

Global Hölder estimates for equations of Monge-Ampère type

Summary We prove interior oscillation and global Hölder estimates, independent of any boundary data, for convex solutions of certain types of Monge-Ampère equations under suitable conditions on the equation and the domain ⁿ . We also deduce the existence, uniqueness, regularity and unboundedness, under suitable conditions, of convex extremal solutions of certain Monge-Ampère equations.     

Monge-Ampère方程边界爆破解的最优估计和不存在性

该文致力于研究如下Monge-Ampère方程边界爆破解的最优估计和严格凸解的不存在性M[u](x)=K(x)f(u),x∈Ω,u(x)→+∞当dist(x,?Ω)→0.这里M[u]=det(u_xixj)是Monge-Ampère算子,Ω是R^N(N≥2)中的光滑有界严格凸区域.文中不仅得到了K(x)和f(u)的各种条件之间的关系,还通过和已有文献中相关结果的比较明确了条件和估计之间的关系.并且,在Ω是一般区域的情况下给出了严格凸解不存在的结果,而这在以往文献中尚未提及.     

一类Monge-Ampère方程的特征值问题

对一类Monge-Amp(e)re方程的特征值问题进行了研究.通过移动平面法证明了在凸对称区域内,Dirichlet问题的C2凹(凸)解一定是对称的.进而通过对常微分方程和椭圆形偏微分方程的讨论,得到一类n维单位球上特征值问题的非平凡解的存在性和正则性结果.     

A REMARK OF THE HLDER ESTIMATE OFSOLUTIONS OF SOME DEGENERATE PARABOLICEQUATIONS

1Intr0ducti0nWeconsiderthedegenerateparabolicequationswherejisasmo0thfunction.F0rgeneraldegenerateparab0licequati0nsoftype(1.1),therearealotofpaperscon-sideringtheeXistence,uniquenessandregUlarity0ftheweaksolutions.Mostoftheavailableresult8havetod0withlocalH5ldercontinuityofsolutionsandregularityupt0theb0undarywithboliciteXPonents.InthispaPer,westudytheH5lderesthoate0fthes0lutionforthedegenerateparab0licequati0nhionedimentionalcase,theekistence(1相似文献   

On the Dirichlet problem for Monge-Ampère type equations

In this paper, we prove second derivative estimates together with classical solvability for the Dirichlet problem of certain Monge-Ampére type equations under sharp hypotheses. In particular we assume that the matrix function in the augmented Hessian is regular in the sense used by Trudinger and Wang in Ann. Scoula Norm. Sup. Pisa Cl. Sci. VIII, 143–174 2009 in their study of global regularity in optimal transportation as well as the existence of a smooth subsolution. The latter hypothesis replaces a barrier condition also used in their work. The applications to optimal transportation and prescribed Jacobian equations are also indicated.     

Global solvability of monge-ampère type equations

We study the global solvability of Monge-Ampère equations of mixed type by \"blowing up\" the problem onto the torus embedded at the singular point of the equations     

Comparison results for Monge-Ampère type equations with lower order terms

In this paper we deal with Monge-Ampère type equations in two dimensions and, using the symmetrization with respect to the perimeter, we prove some comparison results for solutions of such equations involving the solutions of conveniently symmetrized problems.     

一般区域上的抛物型Monge-Ampère方程(Ⅱ)

本文证明了一般光滑区域上抛物型Ｍｏｎｇｅ－Ａｍｐèｒｅ方程第一边值问题古典解的存在性，推广了凸区域上相应的结果。     

Some new estimates for solutions of degenerate two dimensional Monge-Ampère equations

We derive a monotonicity formula for smooth solutions u of degenerate two dimensional Monge-Ampère equations, and use this to obtain a local H?lder gradient estimate, depending on for some . Received August 9, 1999; in final form December 8, 1999/ Published online December 8, 2000