On Harnack’s Inequality for the Linearized Parabolic Monge-Ampère Equation

It is shown that the parabolic Harnack property stands as an intrinsic feature of the Monge-Ampère quasi-metric structure by proving Harnack’s inequality for non-negative solutions to the linearized parabolic Monge-Ampère equation under minimal geometric assumptions.     

The equation of complex Monge-Ampère type and stability of solutions

We prove that in a family of plurisubharmonic functions with Monge-Ampère measures bounded from above by such a measure of one function weak convergence is equivalent to convergence in capacity. We also show a very general statement on the existence of solutions of the complex Monge-Ampère type equation. First author partially supported by the Swedish Research Council contract 621-2002-5308. Second author partially supported by Polish ministerial Grant No. 1 PO3A 037 27.     

Solving the 2-D elliptic Monge-Ampère equation by a Kansa’s method

In this paper, a Kansa’s method is designed to solve numerically the Monge-Ampère equation. The primitive Kansa’s method is a meshfree method which applying the combination of some radial basis functions (such as Hardy’s MQ) to approximate the solution of the linear parabolic, hyperbolic and elliptic problems. But this method is deteriorated when is used to solve nonlinear partial differential equations. We approximate the solution in some local triangular subdomains by using the combination of some cubic polynomials. Then the given problems can be computed in each subdomains independently. We prove the stability and convergence of the new method for the elliptic Monge-Ampère equation. Finally, some numerical experiments are presented to demonstrate the theoretical results.     

Stability of radial symmetry for a Monge-Ampère overdetermined problem

Recently the symmetry of solutions to overdetermined problems has been established for the class of Hessian operators, including the Monge-Ampère operator. In this paper we prove that the radial symmetry of the domain and of the solution to an overdetermined Dirichlet problem for the Monge-Ampère equation is stable under suitable perturbations of the data.      

On the solution of Dirichlet problem of complex Monge-Ampère equation on Cartan-Hartogs domain of the second type

Complex Monge-Ampère equation is a nonlinear equation with high degree, so its solution is very difficult to get. How to get the plurisubharmonic solution of Dirichlet problem of complex Monge-Ampère equation on the Cartan-Hartogs domain of the second type is discussed by using the analytic method in this paper. Firstly, the complex Monge-Ampère equation is reduced to a nonlinear second-order ordinary differential equation (ODE) by using quite different method. Secondly, the solution of the Dirichlet problem is given in semi-explicit formula, and under a special case the exact solution is obtained. These results may be helpful for the numerical method of Dirichlet problem of complex Monge-Ampère equation on the Cartan-Hartogs domain.     

Symmetry Reduction and Exact Solutions of a Hyperbolic Monge-Amp`ere Equation

By means of the classical symmetry method,a hyperbolic Monge-Ampère equation is investigated.The symmetry group is studied and its corresponding group invariant solutions are constructed.Based on the associated vector of the obtained symmetry,the authors construct the group-invariant optimal system of the hyperbolic Monge-Ampère equation,from which two interesting classes of solutions to the hyperbolic Monge-Ampère equation are obtained successfully.     

On the action of the complex Monge-Ampère operator on piecewise linear functions

The action of the mixed complex Monge-Ampère operator (h ₁, …, h _k ) ? dd ^c h ₁ ∧ … ∧ dd ^c h _k on piecewise linear functions h _i is considered. The language of Monge-Ampère operators is used to transfer results on mixed volumes and tropical varieties to a broader context, which arises under the passage from polynomials to exponential sums. In particular, it is proved that the value of the Monge-Ampère operator depends only on the product of the functions h _i .     

Estimate of the gradient of a solution of the Dirichlet problem for Monge-Ampère type equations in a domain with nonsmooth boundary

An estimate of the normal derivative of a viscous solution to the Dirichlet problem for a nonlinear secondorder elliptic equation with the operator of Monge-Ampère type is constructed on the nonsmooth boundary of the domain. Bibliography: 16 titles. Translated fromProblemy Matematicheskogo Analiza, No. 14, 1995, pp. 110–121.     

A Dirichlet principle for the complex Monge-Ampère operator

A solution to a Dirichlet problem for the complex Monge-Ampère operator is characterized as a minimizer of an energy functional. A mutual energy estimate and a generalization of Hölder's inequality is proved. A comparison is made with corresponding results in classical potential theory.     

OPTIMIZATION APPROACH FOR THE MONGE-AMPèRE EQUATION

In this paper, we introduce and study a method for the numerical solution of the elliptic Monge-Ampère equation with Dirichlet boundary conditions. We formulate the Monge-Ampère equation as an optimization problem. The latter involves a Poisson Problem which is solved by the finite element Galerkin method and the minimum is computed by the conjugate gradient algorithm. We also present some numerical experiments.     

Domains of definition of Monge-Ampère operators on compact Kähler manifolds

Let (X, ω) be a compact Kähler manifold. We introduce and study the largest set DMA(X, ω) of ω-plurisubharmonic (psh) functions on which the complex Monge-Ampère operator is well defined. It is much larger than the corresponding local domain of definition, though still a proper subset of the set PSH(X, ω) of all ω-psh functions. We prove that certain twisted Monge-Ampère operators are well defined for all ω-psh functions. As a consequence, any ω-psh function with slightly attenuated singularities has finite weighted Monge-Ampère energy.     

Regularity of the degenerate Monge-Ampère equation on compact Kähler manifolds

We study the C^1,1 and Lipschitz regularity of the solutions of the degenerate complex Monge-Ampère equation on compact K?hler manifolds. In particular, in view of the local regularity for the complex Monge-Ampère equation, the obtained C^1,1 regularity is a generalization of the Yau theorem which deals with the nondegenerate case. Received: 18 April 2002 / Published online: 24 February 2003 Partially supported by KBN Grant #2 P03A 028 19     

Quaternionic Monge-Ampère equation and Calabi problem for HKT-manifolds

A quaternionic version of the Calabi problem on the Monge-Ampère equation is introduced, namely a quaternionic Monge-Ampère equation on a compact hypercomplex manifold with an HKT-metric. The equation is non-linear elliptic of second order. For a hypercomplex manifold with holonomy in SL(n,ℍ), uniqueness (up to a constant) of a solution is proven, aas well as the zero order a priori estimate. The existence of a solution is conjectured, similar to the Calabi-Yau theorem. We reformulate this quaternionic equation as a special case of the complex Hessian equation, making sense on any complex manifold.     

The problem of regular isometric embeddings and the Monge-Ampère equation of hyperbolic type

A relationship between classical approaches to embeddings and the Monge-Ampère equations is described. A new method of constructing smooth solutions of the general Monge-Ampère equation of hyperbolic type for the domains of finite-stripe type is presented. Bibliography: 18 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 234, 1996, pp. 177–186.     

Curvature Estimates for the Level Sets of Solutions to the Monge-Amp`ere Equation detD2u = 1

For the Monge-Ampere equation det D2u = 1, the authors find new auxiliary curvature functions which attain their respective maxima on the boundary. Moreover, the upper bounded estimates for the Gauss curvature and the mean curvature of the level sets for the solution to this equation are obtained.     

Contact Relative Differential Invariants for Non Generic Parabolic Monge-Ampère Equations

We find relative differential invariants of different orders for non generic parabolic Monge-Ampère equations (MAE’s). They are constructed in terms of some tensors associated with the derived flag of the characteristic distribution. The vanishing of such invariants allows one to determine the classes of each non generic parabolic MAE with respect to contact transformations.     

Contact linearization of nondegenerate Monge-Ampère equations

The present paper is devoted to the problem of transforming the classical Monge-Ampère equations to the linear equations by change of variables. The class of Monge-Ampère equations is distinguished from the variety of second-order partial differential equations by the property that this class is closed under contact transformations. This fact was known already to Sophus Lie who studied the Monge-Ampère equations using methods of contact geometry. Therefore it is natural to consider the classification problems for the Monge-Ampère equations with respect to the pseudogroup of contact transformations. In the present paper we give the complete solution to the problem of linearization of regular elliptic and hyperbolic Monge-Ampère equations with respect to contact transformations. In order to solve this problem, we construct invariants of the Monge-Ampère equations and the Laplace differential forms, which involve the classical Laplace invariants as coefficients.     

BMO spaces associated to generalized parabolic sections

Parabolic sections were introduced by Huang[1] to study the parabolic MongeAmpère equation.In this note,we introduce the generalized parabolic sections P and define BMOPq spaces related to these sections.We then establish the John-Nirenberg type inequality and verify that all BMOqP are equivalent for q ≥ 1.     

Notes on Chern��s Affine Bernstein Conjecture

There were two famous conjectures on complete affine maximal surfaces, one due to Calabi, the other to Chern. Both were solved with different methods about one decade ago by studying the associated Euler-Lagrange equation. Here we survey recent developments and techniques in the study of certain Monge-Ampère equations associated with Chern??s Affine Bernstein Conjecture, in particular two of its proofs in our recent monograph (Li et?al. in Affine Bernstein problems and Monge-Ampère equations, 2010). In addition, we add some details of the proofs of auxiliary material that were omitted in (Li et?al. in Affine Bernstein problems and Monge-Ampère equations, 2010). The related background information is provided in the Introduction.     

On the W^{2,1+\varepsilon }-estimates for the Monge-Ampère equation and related real analysis

We build upon the techniques introduced by De Philippis and Figalli regarding $W^{2,1+\varepsilon }$ bounds for the Monge-Ampère operator, to improve the recent $A_\infty $ estimates for $\Vert D^2 \varphi \Vert $ to $A_2$ ones. Also, we prove a $(1,2)-$ Poincaré inequality and weak $(q,p)-$ Poincaré inequalities associated to the Monge-Ampère quasi-metric structure. In turn, these Poincaré inequalities are used to prove Harnack’s inequality for non-negative solutions to the linearized Monge-Ampère under minimal geometric assumptions.