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.

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

Let $Ω\subset \R^n$ be a bounded convex domain and $ϕ\in C(\barΩ)$ be a convex function such that $ϕ$ is sufficiently smooth on $\partialΩ$ and the Monge--Ampère measure $\det D^2ϕ$ is bounded away from zero and infinity in $Ω$. The corresponding linearized Monge--Ampère equation is \[ \trace(ΦD^2 u) =f, \] where $Φ:= \det D^2 ϕ~ (D^2ϕ)^{-1}$ is the matrix of cofactors of $D^2ϕ$. We prove a conjecture in \cite{GT} about the relationship between $L^p$ estimates for $D^2 u$ and the closeness between $\det D^2ϕ$ and one. As a consequence, we obtain interior $W^{2,p}$ estimates for solutions to su…

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。

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

We study interior $C^{1, \al}$ regularity of viscosity solutions of the parabolic Monge-Ampére equation $$u_t = b(x,t) \ddua,$$ with exponent $p >0$ and with coefficients $b$ which are bounded and measurable. We show that when $p$ is less than the critical power $\frac{1}{n-2}$ then solutions become instantly $C^{1, \al}$ in the interior. Also, we prove the same result for any power $p>0$ at those points where either the solution separates from the initial data, or where the initial data is $C^{1, β}$.

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.

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

We prove an interior $W^{2,1}$ estimate for singular solutions to the Monge-Ampere equation, and construct an example to show our results are optimal.

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。

Smoothness of radial solutions to Monge-Ampere equations

We characterize when radial weak solutions to Monge-Ampere equations are smooth. This paper extends previous partial results and also covers Generalized Monge-Ampere equations and infinitely vanishing right hand side.

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

We obtain global $W^{2,p}$ estimates for the Monge-Ampere equation under natural assumptions on the domain and boundary data.

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…

The Monge-Ampère equation and its link to optimal transportation

We survey the (old and new) regularity theory for the Monge-Ampère equation, show its connection to optimal transportation, and describe the regularity properties of a general class of Monge-Ampère type equations arising in that context.

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.

Boundary regularity for solutions to the linearized Monge-Ampère equations

We obtain boundary Holder gradient estimates and regularity for solutions to the linearized Monge-Ampere equations under natural assumptions on the domain, Monge-Ampere measures and boundary data. Our results are affine invariant analogues of the boundary Holder gradient estimates of Krylov.

On regularity of complex Monge-Ampere equation

We shall consider the regularity problem of solutions for complex Monge-Ampere equations. First we prove interior $C^2$ estimates of solutions in a bounded domain for complex Monge-Ampere equation with assumption of certain $L^p$ bound for Laplacian u, and of Lipschitz condition on right hand side. Then we shall construct a family of Pogorelov-type examples for complex Monge-Ampere equation. These examples give generalized entire solutions (as well as viscosity solutions) of complex Monge-Ampere equation $\det(u_{i\bar j})=1.

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.

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

We obtain pointwise $C^{2,α}$ estimates at boundary points for solutions to the Monge-Ampere equation under appropriate local conditions on the right hand side and boundary data.

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

In this paper, we establish global $W^{2,p}$ estimates for solutions to the linearized Monge-Ampère equations under natural assumptions on the domain, Monge-Ampère measures and boundary data. Our estimates are affine invariant analogues of the global $W^{2,p}$ estimates of Winter for fully nonlinear, uniformly elliptic equations, and also linearized counterparts of Savin's global $W^{2,p}$ estimates for the Monge-Ampère equations.

Entire invariant solutions to Monge-Ampere equations

We prove existence and regularity of entire solutions to Monge-Ampere equations invariant under an irreducible action of a compact Lie group.

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.

Non linear elliptic theory and the Monge-Ampere equation

The Monge-Ampere equation, plays a central role in the theory of fully non linear equations. In fact we will like to show how the Monge-Ampere equation, links in some way the ideas comming from the calculus of variations and those of the theory of fully non linear equations.

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

In this paper we study an obstacle problem for Monge-Ampère type functionals, whose Euler-Lagrange equations are a class of fourth order equations, including the affine maximal surface equations and Abreu's equation.

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。

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

In this paper, we establish boundary Hölder gradient estimates for solutions to the linearized Monge-Ampère equations with $L^{p}$ ($n<p\leq\infty$) right hand side and $C^{1,γ}$ boundary values under natural assumptions on the domain, boundary data and the Monge-Ampère measure. These estimates extend our previous boundary regularity results for solutions to the linearized Monge-Ampère equations with bounded right hand side and $C^{1, 1}$ boundary data.

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。

Complex Monge-Ampère equations with oblique boundary value

The existence and regularity of the classical plurisubharmonic solution for complex Monge-Ampère equations subject to the semilinear oblique boundary condition which is C^1 perturbation of the Neumann boundary condition, are proved in the certain strictly pseudoconvex domain in C^n.

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.

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

We consider Monge-Ampere equations with bounded right hand side and we study the geometric properties of sections centered at a boundary point. We prove that under natural boundary conditions such sections are equivalent to ellipsoids.

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

We propose nonlinear parabolic equations of Monge--Ampére (M--A) type that admit regional, single poin, and global blow-up of similarity type. A similar model is derived for a fourth-order M--A flow.

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.

Gevrey regularity of subelliptic Monge-Ampère equations in the plane

In this paper, we establish the Gevrey regularity of solutions for a class of degenerate Monge-Ampère equations in the plane, under the assumption that one principle entry of the Hessian is strictly positive and an appropriately finite type degeneracy.

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.

Comparison Principles for subelliptic equations of Monge-Ampere type

We present two comparison principles for viscosity sub- and supersolutions of Monge-Ampere-type equations associated to a family of vector fields. In particular, we obtain the uniqueness of a viscosity solution to the Dirichlet problem for the equation of prescribed horizontal Gauss curvature in a Carnot group.

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。