Entire invariant solutions to Monge-Ampere equations

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…

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.

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.

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.

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。

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.

A note on interior $W^{2,1+\varepsilon}$ estimates for the Monge-Ampere equation

By a variant of the techniques introduced by the first two authors in [DF] to prove that second derivatives of solutions to the Monge-Ampere equation are locally in $L\log L$, we obtain interior $W^{2,1+\varepsilon}$ estimates.

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.

Monge-Ampère方程内部正则性的新方法

New Approach for Interior Regularity of Monge-Ampère Equations

本文发展了一种积分方法，用于研究Monge-Ampère方程$\det D^2 u = 1$严格凸解的内部正则性，提出了新的证明方法。

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。

Partial regularity for singular solutions to the Monge-Ampere equation

We prove that solutions to the Monge-Ampere inequality $$\det D^2u \geq 1$$ in $\mathbb{R}^n$ are strictly convex away from a singular set of Hausdorff $n-1$ dimensional measure zero. Furthermore, we show this is optimal by constructing solutions to $\det D^2u = 1$ with singular set of Hausdorff dimension as close as we like to $n-1$. As a consequence we obtain $W^{2,1}$ regularity for the Monge-Ampere equation with bounded right hand side and unique continuation for the Monge-Ampere equation with sufficiently regular right hand side.

具有自然边界条件的 Monge-Ampère 方程解梯度的连续性估计

Continuity estimates for the gradient of solutions to the Monge-Ampère equation with nature boundary conditions

本文研究 Monge-Ampère 方程解的一阶导数连续模估计，并证明自然边界条件下其梯度具有最优全局 log-Lipschitz 连续性。

Analyticity of the solutions to degenerate Monge-Ampère equations

This paper is devoted to study the following degenerate Monge-Ampère equation: \begin{eqnarray}\label{ab1} \begin{cases} \det D^2 u=Λ_q (-u)^q \quad \text{in}\quad Ω,\\ u=0 \quad\text{on}\quad \partialΩ\end{cases} \end{eqnarray} for some positive constant $Λ_q$. Suppose $Ω\subset\subset \mathbb R^n$ is uniformly convex and analytic. Then the solution of the degenerate Monge-Ampère equation is analytic in $\barΩ$ provided $q\in \mathbb Z^+$.

Singular structures in solutions to the Monge-Ampère equation with point masses

We construct new examples of Monge-Ampère metrics with polyhedral singular structures, motivated by problems related to the optimal transport of point masses and to mirror symmetry. We also analyze the stability of the singular structures under small perturbations of the data given in the problem under consideration.

Monge--Ampère equation and Bellman optimization of Carleson Embedding Theorems

Monge--Ampère equation plays an important part in Analysis. For example, it is instrumental in mass transport problems. On the other hand, the Bellman function technique appeared recently as a way to consider certain Harmonic Analysis problems as the problems of Stochastic Optimal Control. This brings us to Bellman PDE, which in stochastic setting is often a Monge--Ampère equation or its close relative. We explore the way of solving Monge--Ampère equation by a sort of method of characteristics to find the Bellman function of certain classical Harmonic Analysis problems, and, therefore, of find…

区域各向异性凸性与两个Monge-Ampère方程的边界估计

The Anisotropic Convexity of Domains and the Boundary Estimate for Two Monge-Ampère Equations

本文研究了区域各向异性凸性对两类Monge-Ampère方程边界估计的影响，得到了尖锐边界估计和最优全局Hölder正则性。

Global $C^{2,α}$ estimates for the Monge-Ampère equation on polygonal domains in the plane

We classify global solutions of the Monge-Ampère equation $\det D^2 u=1 $ on the first quadrant in the plane with quadratic boundary data. As an application, we obtain global $C^{2,α}$ estimates for the non-degenerate Monge-Ampère equation in convex polygonal domains in $\mathbb R^2$ provided a globally $C^2$, convex strict subsolution exists.

凸多面体中蒙日-安培方程狄利克雷问题的整体正则性

Global regularity for the Dirichlet problem of Monge-Ampère equation in convex polytopes

该研究针对有界凸多面体中蒙日-安培方程的狄利克雷问题，在存在整体C²凸下解的前提下，给出了整体C²和C²,α凸解存在的精确条件，解决了整体正则性相关问题。

Boundary Hölder gradient estimates for the Monge-Ampère equation

We investigate global Hölder gradient estimates for solutions to the Monge-Ampère equation $$\mathrm{det}\;D^2 u=f\quad\mathrm{in}\;Ω,$$ where the right-hand side $f$ is bounded away from $0$ and $\infty$. We consider two main situations when a) the domain $Ω$ is uniformly convex and b) $Ω$ is flat.

The interior $C^2$ estimate for Monge-Ampere equation in dimension $n=2$

In this paper, we introduce a new auxiliary function, and establish the interior $C^2$ estimate for Monge-Ampere equation in dimension $n =2$, which was firstly proved by Heinz \cite{H59}.

Regularity of free boundary for the Monge-Ampère obstacle problem

In this paper, we prove the regularity of the free boundary in the Monge-Ampère obstacle problem $\det D^2 v= f(y)χ_{\{v>0\}}. $ By duality, the regularity of the free boundary is equivalent to that of the asymptotic cone of the solution to the singular Monge-Ampère equation $\det D^2 u = 1/f (Du)+δ_0$ at the origin. We first establish an asymptotic estimate for the solution $u$ near the singular point, then use a partial Legendre transform to change the Monge-Ampère equation to a singular, fully nonlinear elliptic equation, and establish the regularity of solutions to the singular elliptic eq…

Some multi-valued solutions to Monge-Ampere equations

We construct several types of multi-valued solutions to the Monge-Ampere equation in higher dimensions.

Weak subsolutions to complex Monge-Ampère equations

We compare various notions of weak subsolutions to degenerate complex Monge-Ampère equations, showing that they all coincide. This allows us to give an alternative proof of mixed Monge-Ampère inequalities due to Kolodziej and Dinew.

Energy Functionals for the Parabolic Monge-Ampere Equation

We introduce certain energy functionals to the complex Monge-Ampere equation over a bounded domain with inhomogeneous boundary condition, and use these functionals to show the convergence of the solution to the parabolic Monge-Ampere equation.

Some remarks on contact manifolds, Monge-Ampere equations and solution singularities

We describe some natural relations connecting contact geometry, classical Monge-Ampere equations and theory of singularities of solutions to nonlinear PDEs. They reveal the hidden meaning of Monge-Ampere equations and sheds new light on some aspects of contact geometry.

The Dirichlet problem for Monge-Ampère type equations on Hermitian manifolds

In this paper, we solve the Dirichlet problem for Monge-Ampère type equations for $(n-1)$-plurisubharmonic functions on Hermitian manifolds.

Subsolution theorem and the Dirichlet problem for the quaternionic Monge-Ampere equation

In this paper, the author studies quaternionic Monge-Ampère equations and obtain the existence of the solutions to the Dirichlet problem for such equations in strictly pesudoconvex domains in quaternionic space. The stability and subsolution theorems are established for quaternionic Monge-Ampère equations.

Parabolic Complex Monge-Ampère Type Equations on Closed Hermitian Manifolds

We study the parabolic complex Monge-Ampère type equations on closed Hermitian manfolds. We derive uniform $C^\infty$ {\em a priori} estimates for normalized solutions, and then prove the $C^\infty$ convergence. The result also yields a way to carry out method of continuity for elliptic Monge-Ampére type equations.

A new gradient estimate for the complex Monge-Ampère equation

A gradient estimate for complex Monge-Ampère equations which improves in some respects on known estimates is proved using the ABP maximum principle.

A Liouville Theorem for the Complex Monge-Ampère Equation

In this note, we derive a Liouville theorem for the complex Monge-Ampère equation. Our result states that if the global solution $u$ of the complex Monge-Ampère equation with constant right-hand side differs from a quadratic polynomial solution by $o(\abs{x}^2)$ at infinity, then $u$ is a quadratic polynomial.

Dirichlet problem of complex Monge-Ampère equation near an isolated KLT singularity

We solve the Dirichlet Problem of Monge-Ampère equation near an isolate Klt singularity, which generalizes the result of Eyssidieux-Guedj-Zeriahi \cite{EGZ}, where the Monge-Ampère equation is solved on singular varieties without boundary. As a corollary, we construct solutions to Monge-Ampère equation with isolated singularity on strongly pseudoconvex domain $Ω$ contained in $\mathbb{C}^n$.