Huaiyu Jian, You Li, Xushan Tu
In this paper we shall prove the existence, uniqueness and global H$\ddot{o}$lder continuity for the Dirichlet problem of a class of Monge-Ampère type equations which may be degenerate and singular on the boundary of convex domains. We will establish a relation of the H$\ddot{o}$lder exponent for the solutions with the convexity for the domains.
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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 $q\in \mathbb Z^+$.
We obtain $C^{2,β}$ estimates up to the boundary for solutions to degenerate Monge-Ampère equations of the type $$ \det D^2 u = f~~\text{in}~Ω, \quad \quad ~f\sim \text{dist}^α(\cdot, \partialΩ)~\text{near}~\partialΩ,~α>0. $$ As a consequence we obtain global $C^\infty$ estimates up to the boundary for the eigenfunctions of the Monge-Ampère operator $(\det D^2 u)^{1/n}$ on smooth, bounded, uniformly convex domains in $R^n$.
We introduce the concept of k-strictly convexity to describe the accurate convexity of convex domains some directions of which boundary may be flat. Basing this accurate convexity, we construct sub-solutions the Dirichlet problem for some degenerate-singular Monge-Ampère type equations and prove the sharp boundary estimates for convex viscosity solutions of the problem. As a result, we obtain the optimal global Hölder regularity of the convex viscosity solutions.
We consider degenerate Monge-Ampere equations of the type $$\det D^2 u= f \quad \{in $\Om$}, \quad \quad f \sim \, d_{\p \Om}^α\quad \{near $\p \Om$,}$$ where $d_{\p \Om}$ represents the distance to the boundary of the domain $\Om$ and $α>0$ is a positive power. We obtain $C^2$ estimates at the boundary under natural conditions on the boundary data and the right hand side. Similar estimates in two dimensions were obtained by J.X. Hong, G. Huang and W. Wang.
Liouville theorems for a class of degenerate or singular Monge-Ampère equations
本文研究半空间中右端项可退化或奇异的一类 Monge–Ampère 方程，对其解进行分类，并涵盖 Dirichlet 与 Neumann 两种边值情形。
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。
In this paper, we consider the following nonlinear eigenvalue problem for the Monge-Ampére equation: find a non-negative weakly convex classical solution $f$ satisfying {equation*} {cases} \det D^2 f=f^p \quad &\text{in $Ω$} f=\vp \quad &text{on $\partialΩ$} {cases} {equation*} for a strictly convex smooth domain $Ω\subset\R^2$ and $0<p<2$. When $\{f=0\}$ contains a convex domain, we find a classical solution which is smooth on $\bar{\{f>0\}}$ and whose free boundary $\partial\{f=0\}$ is also smooth.
We construct a counterexample to $W^{2,1}$ regularity for convex solutions to $$\det D^2u \leq 1, \quad u|_{\partial Ω} = 0$$ in two dimensions. We also prove a result on the propagation of singularities in two dimensions that are logarithmically slower than Lipschitz. This generalizes a classical result of Alexandrov and is optimal by example.
In this paper, we study the interior C^{1,1} regularity of viscosity solutions for a degenerate Monge-Ampère type equation \det[D^{2}u-A(x, u, Du)]=B(x, u, Du) when B \geq 0 and B^{\frac{1}{n-1}}\in C^{1,1}(\barΩ\times\mathbb{R}\times \mathbb{R}^n). We prove that u\in C^{1,1}(Ω) under the A3 condition and A3w^+ condition respectively. In the former case, we construct a suitable auxiliary function to obtain uniform {\it a priori} estimates directly. In the latter case, the main argument is to establish the Pogorelov type estimates, which are interesting independently.
By constructing appropriate smooth, possibly non-convex supersolutions, we establish sharp lower bounds near the boundary for the modulus of nontrivial solutions to singular and degenerate Monge-Ampère equations of the form $\det D^2 u =|u|^q$ with zero boundary condition on a bounded domain in $\mathbb{R}^n$. These bounds imply that currently known global Hölder regularity results for these equations are optimal for all $q$ negative, and almost optimal for $0\leq q\leq n-2$. Our study also establishes the optimality of global $C^{\frac{1}{n}}$ regularity for convex solutions to the Monge-Ampè…
We obtain solutions of the nonlinear degenerate parabolic equation \[ \frac{\partial ρ}{\partial t} = {div} \Big\{ρ\nabla c^\star [ \nabla (F^\prime(ρ)+V) ] \Big\} \] as a steepest descent of an energy with respect to a convex cost functional. The method used here is variational. It requires less uniform convexity assumption than that imposed by Alt and Luckhaus in their pioneering work \cite{luckhaus:quasilinear}. In fact, their assumption may fail in our equation. This class of problems includes the Fokker-Planck equation, the porous-medium equation, the fast diffusion equation, and the para…
Studying the (long-term) behavior of the Kähler-Ricci flow on mildly singular varieties, one is naturally lead to study weak solutions of degenerate parabolic complex Monge-Ampére equations. The purpose of this article, the first of a series on this subject, is to develop a viscosity theory for degenerate complex Monge-Ampère flows in domains of $\C^n$.
Degenerate Complex Monge Ampere Equation Part I
该研究解决了自丘成桐奠基性工作以来悬而未决的问题，刻画了Kähler流形上退化复Monge-Ampère方程解的奇异性态。
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.
The regularity theory of the degenerate complex Monge-Ampère equation is studied. The equation is considered on a closed compact Kähler manifold $(M,g)$ with nonnegative orthogonal bisectional curvature of dimension $m$. Given a solution $ϕ$ of the degenerate complex Monge-Ampère equation $\det(g_{i \bar{j}} + ϕ_{i \bar{j}}) = f \det(g_{i \bar{j}})$, it is shown that the Laplacian of $ϕ$ can be controlled by a constant depending on $(M,g)$, $\sup f$, and $\inf_M Δf^{1/(m-1)}$.
We obtain global $W^{2,p}$ estimates for the Monge-Ampere equation under natural assumptions on the domain and boundary data.
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.
In this paper, we give a new proof of a celebrated theorem of Jörgens which states that every classical convex solution of \[ \det\nabla^2 u (x)=1\quad {in} \mathbb{R}^2 \] has to be a second order polynomial. Our arguments do not use complex analysis, and can be applied to establish such Liouville type theorems for solutions of a class of degenerate Monge-Ampère equations. We prove that every convex generalized (or Alexandrov) solution of \[ \det \nabla^2 u(x_1,x_2)=|x_1|^α \quad {in} \mathbb{R}^2, \] where $α>-1$, has to be \[ u(x_1,x_2)= \frac{a}{(α+2)(α+1)}|x_1|^{2+α}+\frac{a b^2}{2}x_1^2 …
We obtain a genuine local $C^2$ estimate for the Monge-Ampère equation in dimension two, by using the partial Legendre transform.
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.
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.
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.
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.
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。
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.
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…
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.
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.
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.
In this paper, we prove a $C^{1,1}$ estimate for solutions of complex Monge-Ampère equations on compact almost Hermitian manifolds. Using this $C^{1,1}$ estimate, we show existence of $C^{1,1}$ solutions to the degenerate Monge-Ampère equations, the corresponding Dirichlet problems and the singular Monge-Ampère equations. We also study the singularities of the pluricomplex Green's function. In addition, the proof of the above $C^{1,1}$ estimate is valid for a kind of complex Monge-Ampère type equations. As a geometric application, we prove the $C^{1,1}$ regularity of geodesics in the space of …
Uniqueness for the Degenerate Monge-Ampère Equation on Arbitrary Bounded Convex Domains
该研究针对任意有界凸区域上的退化Monge-Ampère方程，证明p>n时对应Dirichlet问题至多有一个凸Alexandrov解，p=n时同系数非零解互为正倍数，为该类方程的唯一性问题提供了理论依据。
We prove existence and regularity of entire solutions to Monge-Ampere equations invariant under an irreducible action of a compact Lie group.
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.
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.
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.
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.
Uniform estimates for concave homogeneous complex degenerate elliptic equations comparable to the Monge-Ampère equation
该研究依托Kuo和Trudinger的思路、复Monge-Ampère算子多重位势理论及定制的$L^p$-粘性下解理论，证明了一大类完全非线性退化椭圆复方程强上解的精确一致估计。
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.
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。
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.
Studying the (long-term) behavior of the Kähler-Ricci flow on mildly singular varieties, one is naturally lead to study weak solutions of degenerate parabolic complex Monge-Ampére equations. The purpose of this article, the second of a series on this subject, is to develop a viscosity theory for degenerate complex Monge-Ampére flows on compact Kähler manifolds. Our general theory allows in particular to define and study the (normalized) Kähler-Ricci flow on varieties with canonical singularities, generalizing results of Song and Tian.
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.
Local regularity for concave homogeneous complex degenerate elliptic equations comparable to the Monge-Ampère equation
该研究针对与Monge-Ampère方程可比的复退化非线性椭圆方程，建立了其$W^{2,p}_{\mathrm{loc}}$解的局部正则性结果，并将成果应用于$k$-Monge-Ampère方程，推进了复退化椭圆方程的正则性理论研究。
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.
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.
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.
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.
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
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.
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.
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.
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.
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.
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 $0 \leq δ<1$.
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.
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.
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.
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.
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。
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。
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