Optimality Conditions for Smooth Monge Solutions of the Monge–Kantorovich problem

The Monge–Kantorovich problem (MKP) with given marginals defined on closed domains and a smooth cost function is considered. Conditions are obtained (both necessary ones and sufficient ones) for the optimality of a Monge solution generated by a smooth measure-preserving map . The proofs are based on an optimality criterion for a general MKP in terms of nonemptiness of the sets for special functions on X × X generated by c and f. Also, earlier results by the author are used when considering the above-mentioned nonemptiness conditions for the case of smooth .     

复Monge—Ampere方程的特征值问题

讨论了复空间中强拟凸域上的复Ｍｏｎｇｅ－Ａｍｐｅｒｅ方程的特征值问题，证明了特征值问题解的存在唯一性，并给出了这个特征值与一类复空间中复Ｌａｐｌａｃｅ算子的第一特征值的关系，最后利用特征值及特征函数的存在性讨论了一类复Ｍｏｎｇｅ－Ａｍｐｅｒｅ方程的解的存在性及其分岐。     

Monge–Ampère measures on subvarieties

In this article we address the question whether the complex Monge–Ampère equation is solvable for measures with large singular part. We prove that under some conditions there is no solution when the right-hand side is carried by a smooth subvariety in Cⁿ${\mathbb{C}}^n$ of dimension k<n$k<n$.     

Stochastic Monge–Kantorovich problem and its duality

In this article, we prove the existence of a stochastic optimal transference plan for a stochastic Monge–Kantorovich problem by measurable selection theorem. A stochastic version of Kantorovich duality and the characterization of stochastic optimal transference plan are also established. Moreover, Wasserstein distance between two probability kernels is also discussed.     

复Monge－Ampere方程的特征值问题

讨论了复空间中强拟凸域上的复Ｍｏｎｇｅ－Ａｍｐｅｒｅ方程的特征值问题，证明了特征值问题解的存在唯一性，并给出了这个特征值与一类复空间中复Ｌａｐｌａｃｅ算子的第一特征值的关系，最后利用特征值及特征函数的存在性讨论了一类复Ｍｏｎｇｅ－Ａｍｐｅｒｅ方程的解的存在性及其分歧．     

Monge–Kantorovich Norms on Spaces of Vector Measures

One considers Hilbert space valued measures on the Borel sets of a compact metric space. A natural numerical valued integral of vector valued continuous functions with respect to vector valued measures is defined. Using this integral, different norms (we called them Monge–Kantorovich norm, modified Monge–Kantorovich norm and Hanin norm) on the space of measures are introduced, generalizing the theory of (weak) convergence for probability measures on metric spaces. These norms introduce new (equivalent) metrics on the initial compact metric space.     

Causal transport plans and their Monge–Kantorovich problems

This paper investigates causal optimal transport problems. Within this framework, primal attainments and dual formulations are obtained under standard hypothesis, for the related variational problems. Causal transport plans are intrinsically related to martingales by a preserving property. Specific concretizations yield primal problems equivalent to several classical problems of stochastic control, and of stochastic calculus; trivial filtrations yield usual problems of optimal transport.     

On the Monge–Kantorovich problem with additional linear constraints

The Monge–Kantorovich problem with the following additional constraint is considered: the admissible transportation plan must become zero on a fixed subspace of functions. Different subspaces give rise to different additional conditions on transportation plans. The main results are stated in general form and can be carried over to a number of important special cases. They are also valid for the Monge–Kantorovich problem whose solution is sought for the class of invariant or martingale measures. We formulate and prove a criterion for the existence of an optimal solution, a duality assertion of Kantorovich type, and a necessary geometric condition on the support of the optimal measure similar to the standard condition for c-monotonicity.     

Optimal partial mass transportation and obstacle Monge–Kantorovich equation

Optimal partial mass transport, which is a variant of the optimal transport problem, consists in transporting effectively a prescribed amount of mass from a source to a target. The problem was first studied by Caffarelli and McCann (2010) [6] and Figalli (2010) [12] with a particular attention to the quadratic cost. Our aim here is to study the optimal partial mass transport problem with Finsler distance costs including the Monge cost given by the Euclidian distance. Our approach is different and our results do not follow from previous works. Among our results, we introduce a PDE of Monge–Kantorovich type with a double obstacle to characterize active submeasures, Kantorovich potential and optimal flow for the optimal partial transport problem. This new PDE enables us to study the uniqueness and monotonicity results for the active submeasures. Another interesting issue of our approach is its convenience for numerical analysis and computations that we develop in a separate paper [14] (Igbida and Nguyen, 2018).     

Plurifinely Plurisubharmonic Functions and the Monge Ampère Operator

We will define the Monge-Ampère operator on finite (weakly) plurifinely plurisubharmonic functions in plurifinely open sets U???? ⁿ and show that it defines a positive measure. Ingredients of the proof include a direct proof for bounded strongly plurifinely plurisubharmonic functions, which is based on the fact that such functions can plurifinely locally be written as difference of ordinary plurisubharmonic functions, and an approximation result stating that in the Dirichlet norm weakly plurifinely plurisubharmonic functions are locally limits of plurisubharmonic functions. As a consequence of the latter, weakly plurifinely plurisubharmonic functions are strongly plurifinely plurisubharmonic outside of a pluripolar set.     

Hardy Spaces Associated with Monge–Ampère Equation

The main concern of this paper is to study the boundedness of singular integrals related to the Monge–Ampère equation established by Caffarelli and Gutiérrez. They obtained the \(L^2\) boundedness. Since then the \(L^p, 1<p<\infty \), weak (1,1) and the boundedness for these operators on atomic Hardy space were obtained by several authors. It was well known that the geometric conditions on measures play a crucial role in the theory of the Hardy space. In this paper, we establish the Hardy space \(H^p_{\mathcal F}\) via the Littlewood–Paley theory with the Monge–Ampère measure satisfying the doubling property together with the noncollapsing condition, and show the \(H^p_{\mathcal F}\) boundedness of Monge–Ampère singular integrals. The approach is based on the \(L^2\) theory and the main tool is the discrete Calderón reproducing formula associated with the doubling property only.     

Monge–Ampère equations in big cohomology classes

We define non-pluripolar products of an arbitrary number of closed positive (1, 1)-currents on a compact Kähler manifold X. Given a big (1, 1)-cohomology class α on X (i.e. a class that can be represented by a strictly positive current) and a positive measure μ on X of total mass equal to the volume of α and putting no mass on pluripolar sets, we show that μ can be written in a unique way as the top-degree self-intersection in the non-pluripolar sense of a closed positive current in α. We then extend Kolodziedj’s approach to sup-norm estimates to show that the solution has minimal singularities in the sense of Demailly if μ has L ^1+ε -density with respect to Lebesgue measure. If μ is smooth and positive everywhere, we prove that T is smooth on the ample locus of α provided α is nef. Using a fixed point theorem, we finally explain how to construct singular Kähler–Einstein volume forms with minimal singularities on varieties of general type.     

On a time-depending Monge–Ampère type equation

In this paper, we prove a comparison result between a solution u(x,t)$u(x,t)$, x∈Ω⊂R²$x\in \Omega \subset {\mathbb{R}}^2$, t∈(0,T)$t\in (0,T)$, of a time depending equation involving the Monge–Ampère operator in the plane and the solution of a conveniently symmetrized parabolic equation. To this aim, we prove a derivation formula for the integral of a smooth function g(x,t)$g(x,t)$ over sublevel sets of u$u$, {x∈Ω:u(x,t)<?}$\{x\in \Omega :u(x,t)<?\}$, ?∈R$?\in \mathbb{R}$, having the same perimeter in R²${\mathbb{R}}^2$.     

An inequality for mixed Monge–Ampère measures

We generalize an inequality for mixed Monge–Ampère measures from Kołodziej (Indiana Univ. Math. J. 43, 1321–1338, 1994). We also give an example that shows that our assumptions are sharp. The corresponding result in the setting of compact K?hler manifold is also discussed.     

Monge–Ampère Type Equations on Almost Hermitian Manifolds

In this paper we consider the Monge–Ampère type equations on compact almost Hermitian manifolds. We derive C^∞ a priori estimates under the existence of an admissible C-subsolution. Finally,we obtain an existence result if there exists an admissible supersolution.