This paper studies the pressureless Euler-Poisson system and its fully non-linear counterpart, the Euler-Monge-Ampère system, where the fully non-linear Monge-Ampère equation substitutes for the linear Poisson equation. While the first is a model of plasma physics, the second is derived as a geometric approximation to the Euler incompressible equations. Using energy estimates, convergence of both systems to the Euler incompressible equations is proved.
This paper studies the Vlasov-Monge-Ampere system (VMA), a fully non-linear version of the Vlasov-Poisson system (VP) where the (real) Monge-Ampere equation substitutes for the usual Poisson equation. This system can be derived as a geometric approximation of the Euler equations of incompressible fluid mechanics in the spirit of Arnold and Ebin. Global existence of weak solutions and local existence of smooth solutions are obtained. Links between the VMA system, the VP system and the Euler equations are established through rigorous asymptotic analysis.
We study the global wellposedness of the Euler-Monge-Ampère (EMA) system. We obtain a sharp, explicit critical threshold in the space of initial configurations which guarantees the global regularity of EMA system with radially symmetric initial data. The result is obtained using two independent approaches -- one using spectral dynamics of Liu & Tadmor [Comm. Math. Physics 228(3):435-466, 2002] and another based on the geometric approach of Brenier & Loeper [Geom. Funct. Analysis 14(6):1182--1218, 2004]. The results are extended to 2D radial EMA with swirl.
We identify a novel connection between a recently introduced pseudo-Riemannian framework for optimal mass transport and the geometry of Monge-Ampère equations. We show this correspondence by application to an example from geophysical fluid dynamics.
This paper concerns the questions of flexibility and rigidity of solutions to the Monge-Ampère equation which arises as a natural geometrical constraint in prestrained nonlinear elasticity. In particular, we focus on anomalous i.e. "flexible" weak solutions that can be constructed through methods of convex integration à la Nash & Kuiper and establish the related h-principle for the Monge-Ampère equation in two dimensions
We derive a new model for pre-strained thin films, which consists of minimizing a biharmonic energy of deformations $v\in W^{2,2}$ satisfying the Monge-Ampère constraint $\det\nabla^2v = f$. We further discuss multiplicity properties of the minimizers of this model, in some special cases.
This article is concerned with the question: For which pairs of hyperbolic Euler-Lagrange systems in the plane does there exist a rank-$1$ Bäcklund transformation relating them? We express some obstructions to such existence in terms of the local invariants of the Euler-Lagrange systems. In addition, we discover a class of Bäcklund transformations relating two hyperbolic Euler-Lagrange systems of distinct types.
We describe a method to reduce partial differential equations of Monge-Ampère type in 4 variables to complex partial differential equations in 2 variables. To illustrate this method, we construct explicit holomorphic solutions of the special lagrangian equation, the real Monge-Ampère equations and the Plebanski 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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
We study various capacities on compact Kähler manifolds which generalize the Bedford-Taylor Monge-Ampère capacity. We then use these capacities to study the existence and the regularity of solutions of complex Monge-Ampère equations.
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.
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.
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 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.
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.
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 $\mathbb{C}^n$.
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.
We show the volume maximizing property of the special Lagrangian submanifolds of a pseudo-Euclidean space. These special Lagrangian submanifolds arise locally as gradient graphs of solutions to Monge-Ampere Equations.
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 prove the long time existence and uniqueness of solutions to the parabolic Monge-Ampère equation on compact almost Hermitian manifolds. We also show that the normalization of solution converges to a smooth function in $C^{\infty}$ topology as $t\rightarrow\infty$. Up to scaling, the limit function is a solution of the Monge-Ampère equation. This gives a parabolic proof of existence of solutions to the Monge-Ampère equation on almost Hermitian manifolds.
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 $0 \leq δ<1$.
The fractional nonlocal linearized Monge--Ampère equation is introduced. A Harnack inequality for nonnegative solutions to the Poisson problem on Monge--Ampère sections is proved.
