Sur les equations de Monge-Ampere. I

In this paper we study the real Monge-Arapère equations: det(D²u)= f(x) in 0, u convex in 0, u=0 on 0, and we introduce a new method for solving these equations which enables us to show the existence of regular solutions. This method uses only p.d.e. techniques and does not use any geometrical results. Furthermore, it enables us to solve quasilinear Monge-Ampère equations.     

Global Hölder estimates for equations of Monge-Ampère type

Summary We prove interior oscillation and global Hölder estimates, independent of any boundary data, for convex solutions of certain types of Monge-Ampère equations under suitable conditions on the equation and the domain ⁿ . We also deduce the existence, uniqueness, regularity and unboundedness, under suitable conditions, of convex extremal solutions of certain Monge-Ampère equations.     

An efficient approach for the numerical solution of the Monge-Ampère equation

In this paper, we present a new method to compute the numerical solution of the elliptic Monge-Ampère equation. This method is based on solving a parabolic Monge-Ampère equation for the steady state solution. We study the problem of global existence, uniqueness, and convergence of the solution of the fully nonlinear parabolic PDE to the unique solution of the elliptic Monge-Ampère equation. Some numerical experiments are presented to show the convergence and the regularity of the numerical solution.     

On differential equations characterized by their Lie point symmetries

We study the geometry of differential equations determined uniquely by their point symmetries, that we call Lie remarkable. We determine necessary and sufficient conditions for a differential equation to be Lie remarkable. Furthermore, we see how, in some cases, Lie remarkability is related to the existence of invariant solutions. We apply our results to minimal submanifold equations and to Monge-Ampère equations in two independent variables of various orders.     

On Jörgens, Calabi, and Pogorelov type theorem and isolated singularities of parabolic Monge-Ampère equations

In the paper, we extend Jörgens, Calabi, and Pogorelov's theorem on entire solutions of elliptic Monge-Ampère equations to parabolic equations associated with Gauss curvature flows. Our results include Gutiérrez and Huang's previous work as a special case. Besides, we also treat the isolated singularities for parabolic Monge-Ampère equations that was firstly studied by Jörgens for elliptic case in two dimensions.     

Triple nontrivial radial convex solutions of systems of Monge-Ampère equations

By using the Leggett-Williams fixed point theorem, this paper investigates the existence of at least three nontrivial radial convex solutions of systems of Monge-Ampère equations.     

A note on the contact angle boundary condition for Monge-Ampère equations

We give a simple proof of a result of Xinan Ma concerning a necessary condition for the solvability of the two-dimensional Monge-Ampère equation subject to the contact angle or capillarity boundary condition. Our technique works for more general Monge-Ampère equations in any dimension, and also extends to some other boundary conditions.

     

Second-order type-changing evolution equations with first-order intermediate equations

This paper presents a partial classification for C^∞ type-changing symplectic Monge-Ampère partial differential equations (PDEs) that possess an infinite set of first-order intermediate PDEs. The normal forms will be quasi-linear evolution equations whose types change from hyperbolic to either parabolic or to zero. The zero points can be viewed as analogous to singular points in ordinary differential equations. In some cases, intermediate PDEs can be used to establish existence of solutions for ill-posed initial value problems.     

Convex solutions of boundary value problems

We establish two criteria for the existence of convex solutions for a boundary value problem arising from the study of the existence of convex radial solutions for the Monge-Ampère equations. We shall use fixed point theorems in a cone.     

Regularity estimates of solutions to complex Monge-Ampère equations on Hermitian manifolds

In this paper, we obtain the Bedford-Taylor interior C² estimate and local Calabi C³ estimate for the solutions to complex Monge-Ampère equations on Hermitian manifolds.     

Complex Monge-Ampère equations and totally real submanifolds

We study the Dirichlet problem for complex Monge-Ampère equations in Hermitian manifolds with general (non-pseudoconvex) boundary. Our main result (Theorem 1.1) extends the classical theorem of Caffarelli, Kohn, Nirenberg and Spruck in Cn. We also consider the equation on compact manifolds without boundary, attempting to generalize Yau's theorems in the Kähler case. As applications of the main result we study some connections between the homogeneous complex Monge-Ampère (HCMA) equation and totally real submanifolds, and a special Dirichlet problem for the HCMA equation related to Donaldson's conjecture on geodesics in the space of Kähler metrics.     

Approximation results by difference schemes of fracture energies: the vectorial case

In this paper we deal with Monge-Ampère type equations in two dimensions and, using the symmetrization with respect to the perimeter, we prove some comparison results for solutions of such equations involving the solutions of conveniently symmetrized problems.     

反射天线设计的一个数学理论

本文讨论反射天线面设计中出现的一个偏微分方程, 这是一个完全非线性Monge-Ampère型方程. 我们先给出一个广义解的定义, 然后介绍如何得到广义解的存在性、唯一性和正则性, 最后我们给出求数值解的一个方法.     

Comparison results for Monge-Ampère type equations with lower order terms

In this paper we deal with Monge-Ampère type equations in two dimensions and, using the symmetrization with respect to the perimeter, we prove some comparison results for solutions of such equations involving the solutions of conveniently symmetrized problems.     

The first initial-boundary value problem for fully nonlinear parabolic equations generated by functions of the eigenvalues of the Hessian

For a class of elliptic Hessian operators raised by Caffarelli-Nirenberg-Spruck, the corresponding parabolic Monge-Ampère equation was studied, the existence and uniqueness of the admissible solution to the first initial-boundary value problem for the equation were established, which extended a result of Ivochkina-Ladyzhenskaya.     

A research into the numerical method of Dirichlet's problem of complex Monge-Ampère equation on Cartan-Hartogs domain of the third type

Monge-Ampère equation is a nonlinear equation with high degree, therefore its numerical solution is very important and very difficult. In present paper the numerical method of Dirichlet's problem of Monge-Ampère equation on Cartan-Hartogs domain of the third type is discussed by using the analytic method. Firstly, the Monge-Ampère equation is reduced to the nonlinear ordinary differential equation, then the numerical method of the Dirichlet problem of Monge-Ampère equation becomes the numerical method of two point boundary value problem of the nonlinear ordinary differential equation. Secondly, the solution of the Dirichlet problem is given in explicit formula under the special case, which can be used to check the numerical solution of the Dirichlet problem.     

Nets of lines depending on a parameter and sufficient conditions for their convergence

We study criteria for the global regularity of a net of lines defined on the plane or a part of it by an ordinary differential equation of first order and second degree, nets depending on a parameter, and questions of convergence on the parameter. We use an analytic technique connected with hyperbolic systems of equations of a special form. We give applications to surfaces of negative Gaussian curvature and to hyperbolic Monge-Ampère equations.Translated fromItogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 22, 1990, pp. 3–36.     

Monge-Ampère foliations with singularities at the boundary of strongly convex domains

Let be a bounded strongly convex domain with smooth boundary. We consider a Monge-Ampère type equation in D with a simple pole at the boundary. Using the Lempert foliation of D in extremal discs, we construct a solution u whose level sets are boundaries of horospheres. Among other things, we show that the biholomorphisms between strongly convex domains are exactly those maps which preserves our solution.     

Solution of the Monge-Ampère equation on Wiener space for general log-concave measures

In this work we prove that the unique 1-convex solution of the Monge-Kantorovitch measure transportation problem between the Wiener measure and a target measure which has an H-log-concave density, in the sense of Feyel and Üstünel [J. Funct. Anal. 176 (2000) 400-428], w.r.t the Wiener measure is also the strong solution of the Monge-Ampère equation in the frame of infinite-dimensional Fréchet spaces. We further enhance the polar factorization results of the mappings which transform a spread measure to another one in terms of the measure transportation of Monge-Kantorovitch and clarify the relation between this concept and the Itô-solutions of the Monge-Ampère equation.     

Entire invariant solutions to Monge-Ampère equations

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