Interior $C^2$ estimate for Monge-Ampère equation in dimension two

An Interior A Priori Estimate for Solutions to Monge-Ampère Equations with Right-Hand Side Close to One

We consider Monge-Ampére equations with the right hand side function close to a constant and from a function class that is larger than any Hölder class and smaller than the Dini-continuous class. We establish an upper bound for the modulus of continuity of the solution's second derivatives. This bound depends exponentially on a quantity similar to but larger than the Dini semi-norm. We establish explicit control on the shape of the sequence of shrinking sections, hence revealing the nature of such exponential dependence.

线性化蒙日-安培方程解的内二阶导数估计

Interior second derivative estimates for solutions to the linearized Monge--Ampère equation

设$Ω\subset \R^n$是有界凸域，$ϕ\in C(\barΩ)$是凸函数，使得$ϕ$在$\partialΩ$上是充分光滑的，并且Monge-Ampère测度$\det D^2ϕ$在$Ω$上是远离零和无穷大的有界测度.相应的线性化蒙日-安培方程是\[ \trace(ΦD^2 u) =f, \]，其中$Φ:= \det D^2 ϕ~ (D^2ϕ)^{-1}$是$D^2ϕ$的余因子矩阵。我们证明了一个猜想在\cite{GT}之间的关系$L^p$估计$D^2 u$和$\det D^2ϕ$和一个之间的接近。因此，当测度$\det D^2ϕ$由连续密度给出且函数$f$对某些$q> \max{\{p,n\}}$属于$L^q(Ω)$时，我们得到了该方程解的$W^{2,p}$内估计.

带漂移的散度形式线性化 Monge-Ampère 方程的内部 Harnack 不等式与 Hölder 估计

Interior Harnack inequality and Hölder estimates for linearized Monge-Ampère equations in divergence form with drift

本文研究了带漂移项的散度形式线性化 Monge-Ampère 方程解的内部估计，证明了二维及高维（需附加 Hessian 可积性假设）情形下的内部 Harnack 不等式与 Hölder 估计，推广了 Le 的相关结果。

一类 Monge-Ampère 型方程粘性解的存在性、唯一性与内部正则性

Existence, uniqueness and interior regularity of viscosity solutions for a class of Monge-Ampère type equations

本文研究有界凸区域上可能在边界退化或奇异的 Monge-Ampère 型方程 Dirichlet 问题，通过比较原理构造粘性上下解，并证明了粘性解的存在性、唯一性与内部正则性。

Interior Estimates for Monge-Ampère Equation in Terms of Modulus of Continuity

We investigate the Monge-Ampère equation subject to zero boundary value and with a positive right-hand side unction assumed to be continuous or essentially bounded. Interior estimates of the solution's first and second derivatives are obtained in terms of moduli of continuity. We explicate how the estimates depend on various quantities but have them independent of the solution's modulus of convexity. Our main theorem has many useful consequences. One of them is the nonlinear dependence between the Hölder seminorms of the solution and of the right-side function, which confirms the results of Fi…

线性化Monge-Ampère方程的内部Hölder正则性

Interior Hölder regularity of the linearized Monge-Ampère equation

该研究针对线性化Monge-Ampère方程的内部Hölder正则性问题，二维下用部分Legendre变换给出奇异右端项情形已有结果的新证明并建立新的Moser-Trudinger型不等式，高维下用De Giorgi迭代得到系数可积假设下的内部Hölder估计。

二维线性化Monge–Ampère方程的内部$C^{1,α}$估计

Interior $C^{1,α}$ estimates for the linearized Monge--Ampère equation in two dimensions

该研究针对二维齐次线性化Monge–Ampère方程，在Monge–Ampère密度无连续性假设的条件下，利用部分Legendre变换等方法证明了内部$C^{1,α}$估计，并将其应用于得到具至多线性增长的整体解的Liouville定理。

抛物型Monge-Ampére方程解的$C^{1,\al}$正则性

$C^{1,\al}$ regularity of solutions to parabolic Monge-Ampére equations

研究了抛物型Monge-Ampére方程$$u_t = b(x,t) \ddua,$$黏性解的内部$C^{1, \al}$正则性，其中指数为$p >0$，系数为有界可测的$b$.我们表明，当$p$小于临界功率$\frac{1}{n-2}$，然后解决方案立即成为$C^{1, \al}$在内部。此外，我们证明了相同的结果，任何电源$p>0$在这些点上，无论是从初始数据的解决方案分离，或在初始数据是$C^{1, β}$。

Some interior regularity estimates for solutions of complex Monge-Ampère equations on a ball

In this paper, we consider the Dirichlet problem of a complex Monge-Ampère equation on a ball in $\mathbb C^n$. With $\mathcal C^{1,α}$ (resp. $\mathcal C^{0,α}$) data, we prove an interior $\mathcal C^{1,α}$ (resp. $\mathcal C^{0,α}$) estimate for the solution. These estimates are generalized versions of the Bedford-Taylor interior $\mathcal C^{1,1}$ estimate.

Monge-Ampere方程奇异解的W^{2，1}估计

W^{2,1} estimate for singular solutions to the Monge-Ampere equation

本文证明了Monge-Ampere方程奇异解的$W^{2,1}$内点估计，并构造了一个例子来说明我们的结果是最优的.

线性化 Monge-Ampère 方程的局部势与 Hölder 估计

Local potential and Hölder estimates for the linearized Monge-Ampère equation

本文为右端项为带号测度的线性化 Monge-Ampère 方程建立了局部势与 Hölder 估计，并将其应用于奇异 Abreu 方程的内部估计。

Interior $W^{2,p}$ estimate for small perturbations to the complex Monge-Ampere equation

Let $w_0$ be a bounded, $C^3$, strictly plurisubharmonic function defined on $B_1\subset \mathbb{C}^n$. Then $w_0$ has a neighborhood in $L^{\infty}(B_1)$ with the following property: for any continuous, plurisubharmonic function $u$ in this neighborhood solving $1-\eps \le MA(u)\le 1+\eps$, one has $u\in W^{2,p}(B_{\frac{1}{2}})$, as long as $\eps>0$ is small enough depending only on $n$ and $p$. This partially generalizes Caffarelli's interior $W^{2,p}$ estimates for real Monge-Ampere to the complex version.

The Monge-Ampère Equation

In this survey article we discuss the interior and boundary regularity of Alexandrov solutions to $\det D^2u = 1$. We include some topics which it seems were not recently revisited in similar articles, including Calabi's interior $C^3$ estimate, and the approaches of Cheng-Yau and Lions to obtain classical solutions to the Dirichlet problem. The survey grew from two mini-courses given by the author in May 2018. One was for "Advanced Lectures in Nonlinear Analysis" at l'Università degli Studi di Torino, and the other for the Oxford PDE CDT.

Pogorelov 内部估计与复 Monge-Ampère 方程的 Liouville 定理

Pogorelov interior estimates and a Liouville theorem for the complex Monge--Ampère equation

本文证明复 Monge-Ampère 方程在特定条件下的 Pogorelov 内部估计，并由此推出整体实凸解必为二次多项式，且该结论在外部凸性条件下依然成立。

Monge-Ampere方程径向解的光滑性

Smoothness of radial solutions to Monge-Ampere equations

我们刻画了Monge-Ampere方程的径向弱解何时是光滑的。本文推广了以前的部分结果，也包括了广义Monge-Ampere方程和无穷消失右端。

Monge-Ampere方程的整体$W^{2,p}$估计

Global $W^{2,p}$ estimates for the Monge-Ampere equation

我们获得了全球$W^{2,p}$估计的自然假设下的域和边界数据的Monge-Ampere方程。

Interior Hölder estimate for the linearized complex Monge-Ampere equation

Let $w_0$ be a bounded, $C^3$, strictly plurisubharmonic function defined on $B_1\subset \mathbb{C}^n$. Then $w_0$ has a neighborhood in $L^{\infty}(B_1)$. Suppose that we have a function $ϕ$ in this neighborhood with $1-ε\le MA(u)\le 1+ε$ and there exists a function $u$ solving the linearized complex Monge-Ampere equation: $det(ϕ_{k\bar{l}})ϕ^{I\bar{j}}u_{I\bar{j}}=0$. Then one has an estimate on $|u|_{C^α(B_{\frac{1}{2}})}$ for some $α>0$ depending on $n$, as long as $ε$ is small depending on $n$. This partially generalizes Caffarelli's estimate for linearized real Monge-Ampere equation to t…

蒙日-安培方程及其与最佳运输的联系

The Monge-Ampère equation and its link to optimal transportation

我们调查（旧的和新的）正则性理论的蒙日-安培方程，显示其连接到最佳运输，并描述了正则性的一般类蒙日-安培型方程在这种情况下产生的。

On the Monge-Ampère equation

This text contains the material discussed by the author in the Bourbaki seminar of June 2018, on the recent developments in the theory of the Monge-Ampère equation.

线性化蒙日-安培方程解的边界正则性

Boundary regularity for solutions to the linearized Monge-Ampère equations

我们得到边界持有人梯度估计和正则性的解决方案的线性Monge-Ampere方程的自然假设下的域，Monge-Ampere措施和边界数据。我们的结果是仿射不变的类似物的边界持有人梯度估计的Krylov。

复Monge-Ampere方程的正则性

On regularity of complex Monge-Ampere equation

我们将考虑复Monge-Ampere方程解的正则性问题。首先，在假设Laplacian u有一定的$L^p$界和右边满足利普希茨条件的条件下，证明了复Monge-Ampere方程解在有界区域上的内$C^2$估计.然后我们将构造一族复Monge-Ampere方程的Pogorelov型例子。这些例子给出了复Monge-Ampere方程$ \det（u_{i\bar j}）=1的广义整体解（以及黏性解）.

Asymptotic expansion at infinity of solutions of Monge-Ampère type equations

We obtain a quantitative expansion at infinity of solutions for a kind of Monge-Ampère type equations that origin from mean curvature equations of Lagrangian graph $(x,Du(x))$ and refine the previous study on zero mean curvature equations and the Monge-Ampère equations.

复Monge-Ampère型方程的边界情形

The boundary case for complex Monge-Ampère type equations

该研究聚焦复Monge-Ampère型方程的边界情形问题，在特定几何假设下开展相关分析，为该类方程的边界相关理论研究提供参考。

Boundary Harnack inequality for the linearized Monge-Ampère equations and applications

In this paper, we obtain boundary Harnack estimates and comparison theorem for nonnegative solutions to the linearized Monge-Ampère equations under natural assumptions on the domain, Monge-Ampère measures and boundary data. Our results are boundary versions of Caffarelli and Gutiérrez's interior Harnack inequality for the linearized Monge-Ampère equations. As an application, we obtain sharp upper bound and global $L^p$-integrability for the Green's function of the linearized Monge-Ampère operator.

The eigenvalue problem for the Monge-Ampère operator on general bounded convex domains

In this paper, we study the eigenvalue problem for the Monge-Ampère operator on general bounded convex domains. We prove the existence, uniqueness and variational characterization of the Monge-Ampère eigenvalue. The convex Monge-Ampère eigenfunctions are shown to be unique up to positive multiplicative constants. Our results are the singular counterpart of previous results by P-L. Lions and K. Tso in the smooth, uniformly convex setting. Moreover, we prove the stability of the Monge-Ampère eigenvalue with respect to the Hausdorff convergence of the domains. This stability property makes it pos…

完全非线性椭圆型方程内部估计的新方法

A new approach to interior estimates for fully nonlinear elliptic equations

该研究基于两点最大值原理，通过仅对方程求一次导数的初等方法，证明了Monge-Ampère方程和二次Hessian方程在对应条件下的内部二阶导数估计，拓展了相关方程的估计适用维度。

A Liouville's theorem for some Monge-Ampère type equations

We study a Monge-Ampère type equation that interpolates the classical {σ_2} -Yamabe equation in conformal geometry and the 2-Hessian equation in dimension 4.

Monge-Ampere方程边界点态$C^{2,α}$估计

Pointwise $C^{2,α}$ estimates at the boundary for the Monge-Ampere equation

我们获得逐点$C^{2,α}$估计在边界点的解决方案的蒙赫-安培方程在适当的当地条件下的右手边和边界数据。

线性化蒙日-安培方程解的整体$W^{2,p}$估计

Global $W^{2,p}$ estimates for solutions to the linearized Monge--Ampère equations

本文在关于区域、Monge-Ampère测度和边界数据的自然假设下，建立了线性化蒙日-安培方程解的整体$W^{2,p}$估计.我们的估计是仿射不变的类似物的全球$W^{2,p}$估计的冬季完全非线性，一致椭圆型方程，也线性化对应的Savin的全球$W^{2,p}$估计的蒙日-安培方程。

Monge-Ampere方程的整体不变解

Entire invariant solutions to Monge-Ampere equations

证明了在紧李群的不可约作用下Monge-Ampere方程整体解的存在性和正则性。

A priori estimate for the complex Monge-Ampere equation

In this paper, we prove a Moser-Trudinger type inequality for pluri-subharmonic functions vanishing on the boundary. Our proof uses a descent gradient flow for the complex Monge-Ampere functional.

Moser-Trudinger inequality for the complex Monge-Ampere equation

In this paper, we prove a Moser-Trudinger type inequality for pluri-subharmonic functions vanishing on the boundary. Our proof uses a descent gradient flow for the complex Monge-Ampere functional.

非线性椭圆理论与Monge-Ampere方程

Non linear elliptic theory and the Monge-Ampere equation

蒙日-安培方程在完全非线性方程理论中起着核心作用。事实上，我们将希望显示如何蒙日安培方程，链接在某种程度上的想法来自变分法和理论的充分非线性方程。

一类Monge-Ampère型泛函的障碍问题

An obstacle problem for a class of Monge-Ampère type functionals

本文研究Monge-Ampère型泛函的障碍问题，其Euler-Lagrange方程是一类四阶方程，包括仿射极大曲面方程和Abreu方程.

Optimisation approach for the Monge-Ampere equation

This paper studies the numerical approximation of solution of the Dirichlet problem for the fully nonlinear Monge-Ampere equation. In this approach, we take the advantage of reformulation the Monge-Ampere problem as an optimization problem, to which we associate a well defined functional whose minimum provides us with the solution to the Monge-Ampere problem after resolving a Poisson problem by the finite element Galerkin method. We present some numerical examples, for which a good approximation is obtained in 68 iterations.

锥体与多边形区域中的Monge-Ampère方程

The Monge-Ampère equation in cones and polygonal domains

本文研究锥体中Monge-Ampère方程解的渐近行为，并利用该展开工具分析多边形区域内解的正则性。

线性化蒙日-安培方程解的边界Hölder梯度估计

On boundary Hölder gradient estimates for solutions to the linearized Monge-Ampère equations

本文在对区域、边界数据和Monge-Ampère测度的自然假设下，建立了线性化的蒙日-安培方程解的边界Hölder梯度估计，其中$L^{p}$（$n<p\leq\infty$）在右端，且具有$C^{1,γ}$边界值.这些估计扩展了我们以前的边界正则性结果的解决方案的线性蒙日-安培方程与有界的右手边和$C^{1, 1}$边界数据。

三维空间中2-蒙特卡洛-安普尔方程的利乌维尔型定理

A Liouville-type theorem for $2$-Monge-Ampère equation in dimension three

本文在三维空间中研究了2-蒙特卡洛-安普尔方程的利乌维尔型定理，证明具有二次增长的解必为二次多项式，通过建立凹性不等式和内部C²估计来完成证明。

具有斜边值的复蒙日-安培方程

Complex Monge-Ampère equations with oblique boundary value

证明了复蒙日-安培方程在Neumann边界条件的C^1扰动下，在C^n中的某个严格拟凸区域上古典多重次调和解的存在性和正则性.

A double large deviation principle for monge-ampere gravitation

Monge-Ampere gravitation is a nonlinear modification of classical Newtonian gravitation, when the Monge-Ampere equation substitutes for the Poisson equation. We establish, through two applications of the large deviation principle, that the MA gravitation for a finite number of particles can be reduced, through a double application of the large deviation principle, to the simplest possible stochastic model: a collection of independent Brownian motions with vanishing noise.

Monge-Ampere方程在边界处的一个局部化性质

A localization property at the boundary for the Monge-Ampere equation

本文研究了右端有界的Monge-Ampere方程，并研究了以边界点为中心的截面的几何性质。我们证明了在自然边界条件下，这样的部分是等价的椭球。

Monge-Ampère型抛物型方程的自相似爆破

Self-similar blow-up in parabolic equations of Monge--Ampère type

本文提出了一类非线性Monge-Ampére（M-A）型抛物型方程，该方程允许区域性、单点性和相似型整体爆破.一个类似的模型推导出一个四阶M-A流。

Asymptotics and convergence for the complex Monge-Ampere equation

We study the asymptotics of complete Kaehler-Einstein metrics on strictly pseudoconvex domains in C^n and derive a convergence theorem for solutions to the corresponding Monge-Ampere equation. If only a portion of the boundary is analytic, the solutions satisfy Gevrey type estimates for tangential derivatives. A counterexample for the model linearized equation suggests that there is no local convergence theorem for the complex Monge-Ampere equation

Existence and nonexistence to exterior Dirichlet problem For Monge-Ampère equation

We consider the exterior Dirichlet problem for Monge-Ampère equation with prescribed asymptotic behavior. Based on earlier work by Caffarelli and the first named author, we complete the characterization of the existence and nonexistence of solutions in terms of their asymptotic behaviors.

A non local Monge-Ampere equation

We introduce a non local analog to the Monge-Ampere operator and show some of its properties. We prove that a global problem involving this operator has C 1,1 solutions in the full space.

一种变分方法用于带有混合测度和单调性的退化Monge-AMPERE方程

A Variational Approach to Degenerate Monge--Ampère Equations with Mixed Measures and Monotonicity

本文提出一种变分方法研究退化Monge-AMPERE方程，结合混合测度与单调性，通过能量分析、最大原理和比较原理，揭示了特征值问题在能量类外的解存在性及对称性破坏现象。

Boundary Estimates for solutions of the Monge-Ampere equation satisfying Dirichlet-Neumann type conditions in annular domains

We consider smooth solutions to the Monge-Amp`ere equation subject to mixed boundary conditions on annular domains. We establish global $C^2$ estimates when the boundary of the domain consists of two smooth strictly convex closed hypersurfaces.

The conical complex Monge-Ampère equations on Kähler manifolds

In this paper, by providing the uniform gradient estimates for a sequence of the approximating equations, we prove the existence, uniqueness and regularity of the conical parabolic complex Monge-Ampère equation with weak initial data. As an application, we prove a regularity estimates, that is, any $L^{\infty}$-solution of the conical complex Monge-Ampère equation admits the $C^{2,α,β}$-regularity.

平面上次椭圆型蒙日-安培方程的Gevrey正则性

Gevrey regularity of subelliptic Monge-Ampère equations in the plane

本文在Hessian方程的主项严格为正且具有适当的有限型退化的条件下，建立了平面上一类退化的蒙日-安培方程解的Gevrey正则性.

Elliptic solutions to nonsymmetric Monge-Ampère type equations I. The $d$-concavity and the comparison principle

We introduce the so-called $d$-concavity, $d \geq 0,$ and prove that the nonsymmetric Monge-Ampère type function of matrix variable is concave in an appropriate unbounded and convex set. We prove also the comparison principle for nonsymmetric Monge-Ampère type equations in the case when they are so-called $δ$-elliptic with respect to compared functions with $0 \leq δ<1$.

Global $W^{2,1+\varepsilon}$ estimates for Monge-Ampère equation with natural boundary condition

For the Monge-Ampère equation with a right-hand side bounded away from 0 and infinity, we show that the solution, subject to the natural boundary condition arising in optimal transport, is in $W^{2,1+\varepsilon}$ up to the boundary.

Visualization of the convex integration solutions to the Monge-Ampère equation

In this article, we implement the algorithm based on the convex integration result proved in [Lewicka-Pakzad Analysis and PDE (2017)] and obtain visualizations of the first iterations of the Nash-Kuiper scheme, approx- imating the anomalous solutions to the Monge-Ampère equation in two dimensions.

Monge-Ampere型次椭圆方程的比较原理

Comparison Principles for subelliptic equations of Monge-Ampere type

本文给出了一类向量场的Monge-Ampere型方程的粘性子解和上解的两个比较原理。特别地，我们得到了在卡诺群中具有给定水平高斯曲率的方程的狄利克雷问题的黏性解的唯一性。

具有低共 dimension 突变的 Monge-Ampère 方程解

Solutions to Monge-Ampère Equations with Low Codimensional singularities

本文研究了具有低共 dimension 突变的 Monge-Ampère 方程解的构造及其正则性，探讨了最优传输中潜在函数满足的 Monge-Ampère 方程。