We survey 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.

2 Formulation of the mass transport problems 4 2.1 The original Monge-Kantorovich problem . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Guessing a good dual to the Monge-Kantorovich problem . . . . . . . . . . . . . 6 2.3 Properties of ”Extreme points of C” . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4 Existence of a minimizer . . . . . . . . . . . . . . . . . . . . . . . . . . . ...

Pluri-subharmonic (psh) functions play a primary role in pluri-potential theory. They are closely related to the operator dd c = 2i∂ ¯ ∂ (with notation d = ∂ + ¯ ∂ and d c = i(¯ ∂ − ∂)), which serves as a generalization of the Laplacian from C to C dim for dim > 1. If u is smooth of class C 2 , then for 1 ≤ n ≤ dim, the coefficients of the exterior power (dd c u) n are given by the n×n sub-dete...

Recently a Dynamic-Monge-Kantorovich formulation of the PDE-based -optimal transport problem was presented. The model considers diffusion equation enforcing balance transported masses with time-varying conductivity that evolves proportionally to flux. In this paper we present an extension time derivative grows as power-law flux exponent ?>0. A sub-linear growth (0<?<1) penalizes intensity and p...

We consider an Hamilton-Jacobi equation of the form H(x,Du) = 0 x ∈ Ω ⊂ R , (1) where H(x, p) is assumed Borel measurable and quasi-convex in p. The notion of Monge solution, introduced by Newcomb and Su, is adapted to this setting making use of suitable metric devices. We establish the comparison principle for Monge sub and supersolution, existence and uniqueness for equation (1) coupled with ...

We prove the existence of solutions for the Monge minimization problem, addressed in a metric measure space (X, d,m) enjoying the Riemannian curvature-dimension condition RCD∗(K,N), with N < ∞. For the first marginal measure, we assume that μ0 ≪ m. As a corollary, we obtain that the Monge problem and its relaxed version, the Monge-Kantorovich problem, attain the same minimal value. Moreover we ...

We continue the research on the eeects of Monge structures in the area of combinatorial optimization. We show that three optimization problems become easy if the underlying cost matrix fulllls the Monge property: (A) The balanced max{cut problem, (B) the problem of computing minimum weight binary k-matchings and (C) the computation of longest paths in bipartite, edge-weighted graphs. In all thr...