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.     

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.     

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.     

On the uniform estimate in the Calabi-Yau theorem,Ⅱ

We show that a pluripotential proof of the uniform estimate in the Calabi-Yau theorem works also in the Hermitian case.     

Global regularity for a class of Monge-Ampère type equations

In this paper we are concerned with the global regularity of solutions to the Dirichlet problem for a class of Monge-Ampère type equations.By employing the concept of(a,η) type domain,we emphasize that the boundary regularity depends on the convexity of the domain in nature.The key idea of our proof is to provide more effective global H?lder estimates of convex solutions to the problem based on carefully choosing auxiliary functions and constructing sub-solutions.     

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.     

Symmetry Reduction and Exact Solutions of a Hyperbolic Monge-Amp`ere Equation

By means of the classical symmetry method,a hyperbolic Monge-Ampère equation is investigated.The symmetry group is studied and its corresponding group invariant solutions are constructed.Based on the associated vector of the obtained symmetry,the authors construct the group-invariant optimal system of the hyperbolic Monge-Ampère equation,from which two interesting classes of solutions to the hyperbolic Monge-Ampère equation are obtained successfully.     

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.     

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．     

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.     

关于齐次四元Monge-Ampère方程的一些注记北大核心CSCD

本文考虑了四元数空间Hn中齐次四元Monge-Ampère方程的狄利克雷问题解的正则性.首先,当区域是边界为C1,1的强拟凸域时,作者给出了解的Lipschitz估计.其次,考虑了四元MongeAmpère算子的收敛性.最后,讨论了齐次四元Monge-Ampère方程的粘性次解与F-次调和函数之间的关系.     

A Generalised Monge-Ampere Equation

We consider a generalised complex Monge-Amp6re equation on a compact Kahler manifold and treat it using the method of continuity. For complex surfaces we prove an existence result. We also prove that （for three-folds and a related real PDE in a ball in R3） as long as the Hessian is bounded below by a pre-determined constant （whilst moving along the method of continuity path）, a smooth solution exists. Finally, we prove existence for another real PDE in a 3-ball, which is a local real version of a conjecture of X. X. Chen.     

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.     

一类一般形式的抛物型Monge-Ampbre方程

对于Caffarelli-Nirenberg-Spruck提出的一种更一般形式的椭圆型Monge-AmpéFe算子det(D2u+σ),讨论与之对应的～种抛物型Monge-Ampére方程第一初边值同题.在一定的结构性条件下,利用连续性方法证明了其古典解的存在惟-性.     

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.     

与截面相关的Morrey空间与奇异积分

In this paper, we define the Morrey spaces M_F~(p,q) (Rn) and the Campanato spaces E_F~(p,q) (R~n) associated with a family F of sections and a doubling measure μ, where F is closely related to the Monge-Amp`ere equation. Furthermore, we obtain the boundedness of the Hardy-Littlewood maximal function associated to F, Monge-Amp`ere singular integral operators and fractional integrals on M_F~(p,q)(R~n). We also prove that the Morrey spaces M_F~(p,q) (R~n)and the Campanato spaces E_F~(p,q) (R~n) are equivalent with 1 ≤ q ≤ p ∞.     

Khler-Einstein度量的不存在性与退化型复的Monge-Ampére方程的解

本文部分地解决了田刚的一个部分强Ｃ０－估计的猜测［１０－１２］．同时，还解决了一类退化型复的ＭｏｎｇｅＡｍｐéｒｅ解的存在性．     

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。     

On C~(1 ,α) Regularity for Monge- Ampre Equation

In this paper,we consider the well known problem concerning global regularity ofsolutions of the Dirichlet problems for the Monge-Ampère equations.　Theorem1　LetΩ be a uniformly convex domain in Rn with the boundary Ω∈ C3,1 ,φ∈ C2 ,1 ( Ω ) and letf be a nonnegative function in Rnsuch thatf1 / n∈ C1 ,1 (Rn) . Then thereexistsa unique convex solution u∈C1 ,1 / 4 (Ω)∩C1 ,1 (Ω) of the Dirichletproblemdet D2 u =f(x) ,　　 inΩ (1 )u =φ(x) ,　　 on Ω , (2 )where D2 u=[Diju] …     

COMPLEX MONGE-AMPěRE EQUATIONS ON GENERAL DOMAINS

§ 1　 IntroductionThe complex Monge-Ampère equation is an important topic in the theory of severalcomplex variables,and has been intensively studied in last three decades.See[1 ] for thehistorical account and references therein.There are many results proved forstrongly pseu-doconvex domains.The complex Monge-Ampère equations in weakly pseudoconvex do-mains ornon-pseudoconvex domains are also interesting.For example,letΩ0 Ω1 be twostrongly pseudoconvex domains,an intrinsic norm of homo…