Guido De Philippis, Alessio Figalli
In this paper we prove that a strictly convex Alexandrov solution u of the Monge-Ampère equation, with right hand side bounded away from zero and infinity, is $W_{\rm loc}^{2,1}$. This is obtained by showing higher integrability a-priori estimates for $D^2 u$, namely $D^2 u \in L\log^k L$ for any $k\in \mathbb N$.
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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.
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 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 $\det(u_{i\bar j})=1.
New Approach for Interior Regularity of Monge-Ampère Equations
本文发展了一种积分方法，用于研究Monge-Ampère方程$\det D^2 u = 1$严格凸解的内部正则性，提出了新的证明方法。
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…
Regularity of Free Boundary for Monge-Ampère Obstacle Problem up to the Boundary
本文研究Monge-Ampère障碍问题自由边界直到边界的正则性，发现边界值增长阶影响正则性，且二次增长下高维正则性优于二维。
By constructing explicit supersolutions, we obtain the optimal global Hölder regularity for several singular Monge-Ampère equations on general bounded open convex domains including those related to complete affine hyperbolic spheres, and proper affine hyperspheres. Our analysis reveals that certain singular-looking equations, such as $ \det D^2 u = |u|^{-n-2-k} (x\cdot Du -u)^{-k} $ with zero boundary data, have unexpected degenerate nature.
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。
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.
In this note we prove that, if the cost function satisfies some necessary structural conditions and the densities are bounded away from zero and infinity, then strictly $c$-convex potentials arising in optimal transportation belong to $W^{2,1+κ}_{\rm loc}$ for some $κ>0$. This generalizes some recents results concerning the regularity of strictly convex Alexandrov solutions of the Monge-Ampère equation with right hand side bounded away from zero and infinity.
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.
Global $C^{1,α}$ regularity for Monge-Ampère equations on planar convex domains
本文研究了非一致凸域上Monge-Ampère方程的Dirichlet问题，建立了其解的全局Hölder梯度估计，从而证明了全局$C^{1,\alpha}$正则性。
In this paper, we establish interior $C^{1,α}$ estimates for solutions of the linearized Monge-Amp$\grave{e}$re equation $$\mathcal{L}_ϕu:=\mathrm{tr}[ΦD^2 u]=f,$$ where the density of the Monge-Amp$\grave{e}$re measure $g:=\mathrm{det}D^2ϕ$ satisfies a $\mathrm{VMO}$-type condition and $Φ:=(\mathrm{det}D^2ϕ)(D^2ϕ)^{-1}$ is the cofactor matrix of $D^2ϕ$.
In this paper, we establish global $W^{2,p}$ estimates for solutions of the linearized Monge-Amp$\grave{e}$re equation $$\mathcal{L}_ϕu:=\mathrm{tr}[ΦD^2 u]=f,$$ where the density of the Monge-Amp$\grave{e}$re measure $g:=\mathrm{det}D^2ϕ$ satisfies a $\mathrm{VMO}$-type condition, and $Φ:=(\mathrm{det}D^2ϕ)(D^2ϕ)^{-1}$ is the cofactor matrix of $D^2ϕ$.
In this paper, we establish the global $C^{2,α}$ and $W^{2,p}$ regularity for the Monge-Ampère equation $\det\,D^2u = f$ subject to boundary condition $Du(Ω) = Ω^*$, where $Ω$ and $Ω^*$ are bounded convex domains in the Euclidean space $\mathbb{R}^n$ with $C^{1,1}$ boundaries, and $f$ is a Hölder continuous function. This boundary value problem arises naturally in optimal transportation and many other applications.
We consider Monge-Ampère equations with right hand side $f$ that degenerate to $\infty$ near the boundary of a convex domain $Ω$, which are of the type $$\mathrm{det}\;D^2 u=f\quad\mathrm{in}\;Ω,\quad\quad f\sim d^{-α}_{\partialΩ}\quad\mathrm{near}\;\partialΩ,$$ where $d_{\partialΩ}$ represents the distance to $\partial Ω$ and $-α$ is a negative power with $α\in(0,2)$. We study the boundary regularity of the solutions and establish a localization theorem for boundary sections.
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 establish global $C^{1+α,\frac{1+α}{2}}$ estimates for solutions of the linearized parabolic Monge-Amp$\grave{e}$re equation $$\mathcal{L}_ϕu(x,t):=-u_t\,\mathrm{det}D^2ϕ(x)+\mathrm{tr}[Φ(x) D^2 u]=f(x,t)$$ under appropriate conditions on the domain, Monge-Amp$\grave{e}$re measures, boundary data and $f$, where $Φ:=\mathrm{det}(D^2ϕ)(D^2ϕ)^{-1}$ is the cofactor of the Hessian of $D^2ϕ$.
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, 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.
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 $C^{1, β}$.
In this paper, we study the regularity of optimal mappings in Monge's mass transfer problem. Using the approximation to Monge's cost function given by the Euclidean distance c(x,y)=dist(x,y) through the costs c_\eps(x,y)=(\eps^2+dist(x,y)^2)^{1/2}, we consider the optimal mappings T_\eps for these costs, and we prove that the eigenvalues of the Jacobian matrix DT_\eps, which are all positive, are locally uniformly bounded. By an example we prove that T_\eps is in general not uniformly Lipschitz continuous as \eps-0, even if the mass distributions are positive and smooth, and the domains are c-…
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.
In this paper, we introduce an iteration argument to prove that a convex solution to the Monge-Ampère equation $\mbox{det } D^2 u =f $ in dimension two subject to the natural boundary condition $Du(Ω) = Ω^*$ is $C^{2,α}$ smooth up to the boundary. We establish the estimate under the sharp conditions that the inhomogeneous term $f\in C^α$ and the domains are convex and $C^{1,α}$ smooth. When $f\in C^0$ (resp. $1/C<f<C$ for some positive constant $C$), we also obtain the global $W^{2,p}$ (resp. $W^{2,1+ε}$) regularity.
We obtain the Hölder regularity of time derivative of solutions to the dual semigeostrophic equations in two dimensions when the initial potential density is bounded away from zero and infinity. Our main tool is an interior Hölder estimate in two dimensions for an inhomogeneous linearized Monge-Ampère equation with right hand side being the divergence of a bounded vector field. As a further application of our Hölder estimate, we prove the Hölder regularity of the polar factorization for time-dependent maps in two dimensions with densities bounded away from zero and infinity. Our applications i…
Let $μ= e^{-V} \ dx$ be a probability measure and $T = \nabla Φ$ be the optimal transportation mapping pushing forward $μ$ onto a log-concave compactly supported measure $ν= e^{-W} \ dx$. In this paper, we introduce a new approach to the regularity problem for the corresponding Monge--Amp{è}re equation $e^{-V} = \det D^2 Φ\cdot e^{-W(\nabla Φ)}$ in the Besov spaces $W^{γ,1}_{loc}$. We prove that $D^2 Φ\in W^{γ,1}_{loc}$ provided $e^{-V}$ belongs to a proper Besov class and $W$ is convex. In particular, $D^2 Φ\in L^p_{loc}$ for some $p>1$. Our proof does not rely on the previously known regular…
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.
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。
Regularity for the Monge-Ampère equation by doubling
本文利用极大Lagrangian子流形上Hessian的倍增不等式，为Monge-Ampère方程严格凸解的内部正则性提供了一个新证明。
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。
We prove the smoothness of weak solutions to an elliptic complex Monge-Ampere equation, using the smoothing property of the corresponding parabolic flow.
Boundary $C^{2, α}$ Regularity for the Oblique Boundary Value Problem of Monge-Ampère Equations
本文研究Monge-Ampère方程斜边值问题凸解的边界正则性，证明二维情形下的全局 $C^{2,α}$ 估计，并在高维附加切向二次上界条件下推广该估计，同时获得Robin斜边界条件下凸解的存在性。
The Monge-Ampere system in dimension two: a further regularity improvement
本文研究二维任意余维的Monge-Ampère系统，利用Källen方法及迭代Cao-Hirsch-Inauen方法，将凸积分构造的正则性提升至k≥4时的C^{1,1}及一般k时的更高Hölder指数。
The Monge-Ampere system: convex integration with improved regularity in dimension two and arbitrary codimension
该研究针对二维任意余维的Monge-Ampere系统开展凸积分研究，将Hölder正则性柔性指标从C^{1,1/(1+6/k)}提升至C^{1,1/(1+4/k)}，结果与已有相关经典结论一致。
Sobolev inequalities and regularity of the linearized complex Monge-Ampere and Hessian equations
本文研究复$k$-Hessian上下有界时线性化复Monge-Ampère与Hessian方程的正则性，通过建立Green函数估计和新Sobolev不等式，利用Moser迭代得到先验估计与Harnack不等式，适用于实与复情形。
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。
In this note we revisit previous Pogorelov type interior and global second derivative estimates of the author, F. Jiang and J. Liu for solutions of Monge-Amp`ere type partial differential equations. Taking account of recent strict convexity regularity results of Guillen, Kitagawa and Rankin, and following our earlier work in the optimal transportation case, we remove the monotonicity assumptions in the more general case of generated Jacobian equations and consequently in the subsequent application to classical solvability and global regularity for second boundary value problems.
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。
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 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 …
We obtain global $W^{2,p}$ estimates for the Monge-Ampere equation under natural assumptions on the domain and boundary data.
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 present a proof of strict $g$-convexity in 2D for solutions of generated Jacobian equations with a $g$-Monge-Ampère measure bounded away from 0. Subsequently this implies $C^1$ differentiability in the case of a $g$-Monge-Ampère measure bounded from above. Our proof follows one given by Trudinger and Wang in the Monge-Ampère case. Thus, like theirs, our argument is local and yields a quantitative estimate on the $g$-convexity. As a result our differentiability result is new even in the optimal transport case: we weaken previously required domain convexity conditions. Moreover in the optimal…
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.
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。
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.
We provide counterexamples to regularity of optimal maps in the classical Monge problem under various assumptions on the initial data. Our construction is based on a variant of the counterexample in \cite{LSW} to Lipschitz regularity of the monotone optimal map between smooth densities supported on convex domains.
Global $C^{1,β}$ and $W^{2, p}$ regularity for some singular Monge-Ampère equations
本文研究了奇异 Monge-Ampère 方程的全局 $C^{1,β}$ 和 $W^{2, p}$ 正则性，证明了在合适边界数据和域条件下，凸 Aleksandrov 解属于 $C^{1,β}$ 且属于 $W^{2, p}$。
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 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.
Regularity and classification of the free boundary for a Monge-Ampère obstacle problem
本文研究源自$L_p$ Minkowski问题的Monge-Ampère障碍问题，证明了自由边界严格凸部分的$C^{1,\alpha}$正则性及Schauder型估计，并利用Liouville定理给出了$q=0$情形下整体解的完整分类。
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。
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.
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