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(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.     

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.     

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.     

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.     

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．     

Regularity Properties of the Degenerate Monge-Amp`ere Equations on Compact K¨ahler Manifolds

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

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.     

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.     

The Third Initial-boundary Value Problem for a Class of Parabolic Monge-Ampère Equations

For the more general parabolic Monge-Ampère equations defined by theoperator $F(D^2u + σ(x))$, the existence and uniqueness of the admissible solution tothe third initial-boundary value problem for the equation are established. A newstructure condition which is used to get a priori estimate is established.     

The Third Initial-boundary Value Problem for a Class of Parabolic Monge-Ampère Equations

For the more general parabolic Monge-Ampère equations defined by the operator F(D2u + σ(x)),the existence and uniqueness of the admissible solution to the third initial-boundary value problem for the e...     

On a Class of Solutions to the Generalized Derivative Schrödinger Equations

In this work we shall consider the initial value problem associated to the generalized derivative Schrödinger (gDNLS) equationsandFollowing the argument introduced by Cazenave and Naumkin we shall establish the local well-posedness for a class of small data in an appropriate weighted Sobolev space. The other main tools in the proof include the homogeneous and inhomogeneous versions of the Kato smoothing effect for the linear Schrödinger equation established by Kenig-Ponce-Vega.     

Oblique Derivative Problem for Quasilinear Mixed (Lavrentév-Bitsadze) Equations of Second Order in Two Connected Domains

In this article, we first give the representation of solutions for the oblique derivative problem of mixed (Lavrentév-Bitsadze) equations in two connected domains, afterwards prove the uniqueness of solutions of the above problem. Moreover, we prove the solvability of oblique derivative problem for quasilinear mixed (Lavrentév-Bitsadze) equations of second order, and obtain a priori estimates of solutions of the above problem. The above problem is an open problem proposed by Rassias.     

Boundary Behavior of Large Solutions to the Monge-Ampère Equation in a Borderline Case

This paper is concerned with the boundary behavior of strictly convex large solutions to the Monge-Ampère equation detD²u(x)=b(x)f(u(x)), u > 0, x ∈ Ω, where Ω is a strictly convex and bounded smooth domain in R^N with N ≥ 2, f is normalized regularly varying at infinity with the critical index N and has a lower term, and b ∈ C^∞(Ω) is positive in Ω, but may be appropriate singular on the boundary.     

On the Dirichlet Problem for a Class of Singular Complex Monge–Ampère Equations

We study the Dirichlet problem of the n-dimensional complex Monge–Ampère equation det(uij) = F/|z|~(2α), where 0 α n. This equation comes from La Nave–Tian's continuity approach to the Analytic Minimal Model Program.     

The Smoothness of Weak Solutions to the System of Second Order Differential Equations with Non—negative Characteristics

In this paper,we will discuss smoothness of weak solutions for the system of second orderdifferential equations eith non-negative characteristies.First of all,we establish boundary,and interiorestimates and then we prove that solutions of regularization problem satisfy Lipschitz condition.     

A Generalized （G＇/G）-Expansion Method to Find the Traveling Wave Solutions of Nonlinear Evolution Equations

In this article, we construct the exact traveling wave solutions for nonlinear evolution equations in the mathematical physics via the modified Kawahara equation, the nonlinear coupled KdV equations and the classical Boussinesq equations, by using a generalized （G＇/G）-expansion method, where G satisfies the Jacobi elliptic equation. Many exact solutions in terms of Jacobi elliptic functions are obtained.