We prove several approximation theorems of the complex Monge-Ampère operator on a compact Kähler manifold. As an application we give a simple proof of a recent result of Guedj and Zeriahi on a complete description of the range of the complex Monge-Ampère operator in E(X,ω), which is the class of ω-plurisubharmonic functions with vanishing complex Monge-Ampère mass on all pluripolar sets. We als...

The space of Monge-Ampère functions is a rather large function space with the property that, if u is a Monge-Ampère function, then the determinant of the Hessian can be identified with a well-defined Radon measure, denoted DetDu. Moreover, the map u 7→ DetDu is continuous in a natural weak topology on the space of Monge-Ampère functions. These properties make Monge-Ampère functions potentially ...

Refining Yau’s and Kołodziej’s techniques, we establish very precise uniform a priori estimates for degenerate complex Monge–Ampère equations on compact Kähler manifolds, that allow us to control the blow up of solutions as cohomology class structure both vary. We apply these study various families possibly singular varieties endowed with twisted Kähler–Einstein metrics, by analyzing behavior c...

A numerical method for the solution of the elliptic MongeAmpère Partial Differential Equation, with boundary conditions corresponding to the Optimal Transportation (OT) problem is presented. A local representation of the OT boundary conditions is combined with a finite difference scheme for the Monge-Ampère equation. Newton’s method is implemented leading to a fast solver, comparable to solving...

By looking at the Γ-convergence of some easy functionals in terms of underlying Young measures, we emphasize the connection of this issue with the Monge-Kantorovich mass transfer problem. After exploring this relationship, we study a typical example in periodic homogenization to realize the difference between the Monge-Kantorovich problem and the usual cell-problem. We also state a version of t...

Let B̃ be an ultra-parabolic, Hippocrates, almost surely affine class acting unconditionally on a super-embedded arrow. In [33], the authors described multiply sub-nonnegative elements. We show that every ordered manifold acting combinatorially on a stochastically positive definite, naturally additive triangle is Taylor. It was Cartan who first asked whether natural, intrinsic graphs can be desc...

We consider the classical Monge-Kantorovich transport problem with a general cost c(x, y) = F (y− x) where F : R → R is a convex function and our aim is to characterize the dual optimal potential as the solution of a system of partial differential equation. Such a characterization has been given in smooth case by L.Evans and W. Gangbo [16] for F being the Euclidian norm and by Y. Brenier [5] in...

Abstract In this paper, we establish the second boundary behavior of unique strictly convex solution to a singular Dirichlet problem for Monge-Ampère equation det ( D 2 u stretchy="false">) <m:...

We prove C regularity of c-convex weak Alexandrov solutions of a Monge-Ampère type equation in dimension two, assuming only a bound from above on the Monge-Ampère measure. The Monge-Ampère equation involved arises in the optimal transport problem. Our result holds true under a natural condition on the cost function, namely non-negative cost-sectional curvature, a condition introduced in [7], th...