Very weak solutions to the two-dimensional Monge-Ampére equation Dedicated to Professor Jean-Yves Chemin on the Occasion of His 60th Birthday

In this short note we revisit the convex integration approach to constructing very weak solutions to the 2D Monge-Ampére equation with H?lder-continuous ?rst derivatives of exponent β 1/5. Our approach is based on combining the approach of Lewicka and Pakzad(2017) with a new diagonalization procedure which avoids the use of conformal coordinates, which was introduced by De Lellis et al.(2018) for the isometric immersion problem.     

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.     

OPTIMIZATION APPROACH FOR THE MONGE-AMPèRE EQUATION

In this paper, we introduce and study a method for the numerical solution of the elliptic Monge-Ampère equation with Dirichlet boundary conditions. We formulate the Monge-Ampère equation as an optimization problem. The latter involves a Poisson Problem which is solved by the finite element Galerkin method and the minimum is computed by the conjugate gradient algorithm. We also present some numerical experiments.     

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.     

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.     

抛物型Monge-Ampère型方程的Cauchy-Neumann问题

本文研究了一类抛物型Monge-Ampère型方程的Cauchy-Neumann问题.通过构造辅助函数,利用函数在极大值点的性质及柯西不等式等方法对方程的解进行估计,得到了方程解的全局二阶梯度估计.接着利用抛物方程的一般理论,进一步得到在光滑条件下,解的长时间存在性,推广了抛物型Monge-Ampère方程的结果.     

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.     

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.     

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?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、C~(1,α)估计和C~(2,α)估计,以及其在Sasakian几何中的一些应用.这些结果是与S.Dinew、管鹏飞、李超、李嘉禹和张享文等合作得到的.     

Symmetry Reduction and Exact Solutions of a Hyperbolic Monge-Ampère 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 a...     

ON THE EXISTENCE OF SOLUTIONS TO A BI-PLANAR MONGE-AMPèRE EQUATION

In this article, we consider a fully nonlinear partial differential equation which can be expressed as a sum of two Monge-Ampère operators acting in different two-dimensional coordinate sections. This equation is elliptic, for example, in the class of convex functions.We show that the notion of Monge-Ampère measures and Aleksandrov generalized solutions extends to this equation, subject to a weaker notion of convexity which we call bi-planar convexity. While the equation is also elliptic in the class of bi-planar convex functions, the contrary is not necessarily true. This is a substantial difference compared to the classical Monge-Ampère equation where ellipticity and convexity coincide. We provide explicit counter-examples: classical solutions to the bi-planar equation that satisfy the ellipticity condition but are not generalized solutions in the sense introduced. We conclude that the concept of generalized solutions based on convexity arguments is not a natural setting for the bi-planar equation.     

Cartan-Hartogs域的K(a)hler-Einstein度量

研究以不可约有界对称域Ω为底空间的一类Hartogs域Ω上的K(a)ler-Einstein度量,这种域称之为Cartan-Hartogs域,是华罗庚域的一种,其中K(a)ler-Einstein度量的生成函数满足一带有边界条件的复Monge-Ampère方程.一般地,域Ω是非齐性域,其上有一全纯自同构子群以及群不变轨道X∈[0,1],因此可以把复Monge-Ampère方程化为常微分方程,并且此方程在临界值μ0=μ时能够显式解出.临界值μ0对于研究其他不变度量如Bergman度量也是非常有意义的.文中还给出一个猜想,并且证明了该猜想对于两类两类例外域是成立的.     

外球区域Monge-Ampère方程解的对称性（英文）

本文研究了外球区域中一类Monge-Ampère方程解的对称性.利用移动平面法和简-汪引进的一类变换,证明了解是旋转对称的.     

极小曲面方程Bernstein定理的一个推广

本文在一个平面有界集合外部考虑极小曲面方程,借助Monge-Ampère方程的性质,得到了解在无穷远处的渐近行为,扩展了经典的Bernstein定理.     

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.     

正定自伴算子分数幂的泛函不等式

本文研究了super-Poincaré不等式在变换下的稳定性.利用自伴算子的谱分解性质,证明了当正定自伴算子L满足super-Poincaré不等式时,L的分数幂Lα也满足相应的super-Poincaré不等式,并讨论了相应半群的几种超有界性之间的关系.     

带两个形状参数的同次Bézier曲线

构造了一种带两个形状参数的Bézier型曲线,并研究了该曲线的性质、形状参数对曲线的影响及曲线的拼接.所提出的曲线是多项式Bezier曲线的一种同次新扩展,不仅具有传统Bézier曲线的诸多性质,而且可通过修改两个形状参数的取值对其形状进行调节.由于所提出的曲线是一种带有形状参数且与传统Bézier曲线具有相似性质的同次多项式模型,因此比现有的一些带形状参数的Bézier型曲线更有优势.     

Some new estimates for the complex Monge-Ampère equation

In this paper, we consider the complex Monge-Ampère equation posed on a compact K?hler manifold. We show how to get L~p(p ∞) and L∞estimates for the gradient of the solution in terms of the continuity of the right-hand side.     

二维Monge-Ampère型方程的Neumann问题

该文通过构造闸函数将整体约化到边界,证明了二维Monge-Ampère型方程Neumann边值问题解的二阶导数估计,进而得到该方程Neumann边值问题经典解的存在性以及正则性.     

A boundary expansion of solutions to nonlinear singular elliptic equations

In this paper we establish an asymptotic expansion near the boundary for solutions to the Dirichlet problem of elliptic equations with singularities near the boundary.This expansion formula shows the singularity profile of solutions at the boundary.We deal with both linear and nonlinear elliptic equations,including fully nonlinear elliptic equations and equations of Monge-Ampère type.