Isolated singularities of Monge-Ampère equations

Letz=z(x, y) be a real-valued twice continuously differentiable solution of the elliptic Monge-Ampère equationAr+2Bs+Ct+rt – s ²=E in the punctured disk 0<(x–x ₀)²+(y–y ₀)²<². Assume thatq is continuous at (x₀, y₀). Our aim is to give sufficient conditions on the coefficientsA,..., E which ensure that the singularity (x ₀,y ₀) is removable. This generalizes an earlier result of Jörgens (Math. Ann. 129 (1955), 330–344).     

Quaternionic Monge-Ampère equations

The main result of this article is the existence and uniqueness of the solution of the Dirichlet problem for quaternionic Monge-Ampère equations in quaternionic strictly pseudoconvex bounded domains in ℍ ⁿ . We continue the study of the theory of plurisubharmonic functions of quaternionic variables started by the author at [2].     

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.     

A variational approach to complex Monge-Ampère equations

We show that degenerate complex Monge-Ampère equations in a big cohomology class of a compact Kähler manifold can be solved using a variational method, without relying on Yau’s theorem. Our formulation yields in particular a natural pluricomplex analogue of the classical logarithmic energy of a measure. We also investigate Kähler-Einstein equations on Fano manifolds. Using continuous geodesics in the closure of the space of Kähler metrics and Berndtsson’s positivity of direct images, we extend Ding-Tian’s variational characterization and Bando-Mabuchi’s uniqueness result to singular Kähler-Einstein metrics. Finally, using our variational characterization we prove the existence, uniqueness and convergence as k→∞ of k-balanced metrics in the sense of Donaldson both in the (anti)canonical case and with respect to a measure of finite pluricomplex energy.     

Bifurcation for Monge-Ampère equations on flat tori

In [4] (Theorem 2), we solved a Monge-Ampère equation, on a compact Riemannian manifold, which isnot a priori locally invertible. In particular, one cannot expect the solution to be unique. We investigate here, the way bifurcation phenomena may occur in such a case, with some explicit computations on flat tori.     

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.     

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.     

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     

Approximation of solutions of the Monge-Ampère equations by surfaces reduced to developable surfaces

We consider an approximate construction of the surface S that is the graph of a C ²-smooth solution z = z(x, y) of the parabolic Monge-Ampère equation

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.     

The Boundedness of Monge-Ampère Singular Integral Operators

We first define molecules for Hardy spaces H¹_F(\mathbbRⁿ)H^{1}_{\mathcal{F}}(\mathbb{R}^{n}) associated with a family F\mathcal{F} of sections which is closely related to the Monge-Ampère equation and prove their molecular characters. As an application, we show that Monge-Ampère singular operators are bounded on H¹_F(\mathbbRⁿ)H^{1}_{\mathcal{F}}(\mathbb{R}^{n}).     

Singular Solutions to Monge-Ampère Equation

We construct merely Lipschitz and $C^{1,α}$ with rational $α ∈ (0, 1 − 2/n]$ viscosity solutions to the Monge-Ampère equation with constant right hand side.     

拋物型Monge-Ampère方程

<正>这篇文章研究的是方程 H_r+2K_s+L_t+N(rt-s~2)=0.这是一个特殊形式的二阶非线性方程,称为Monge-Ampère方程(简写为 M-A 方程),有椭圆、双曲和抛物型之分.已有不少人研究过椭圆和双曲型的情况,这篇文章试图对抛物型 M-A 方程作-些研究.一般的二阶非线性方程     

The explicit general solution of trivial Monge-Ampère equation

The general solution of the equation with minimal smoothness requirements is presented in explicit form; it depends on 2 functions of one variable. In particular, it allows to describe explicitly all developable surfaces (without planar points) in . The domain and singularities of the solution are investigated. Received: January 5, 1998     

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     

A Class of Singular Coupled Systems of Superlinear Monge-Ampère Equations

Acta Mathematicae Applicatae Sinica, English Series - In this paper, we analyze the existence, multiplicity and nonexistence of nontrivial radial convex solutions to the following system coupled by...     

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.