A Mixed Formulation of the Monge-Kantorovich Equations

On Fluid mechanics formulation of Monge-Kantorovich Mass Transfer Problem

The Monge-Kantorovich mass transfer problem is equivalently formulated as an optimal control prblem for the mass transport equation. The equivalency of the two problems is establish using the Lax-Hopf formula and the optimal control theory arguments. Also, it is shown that the optimal solution to the equivalent control problem is given in a gradient form in terms of the potential solution to th...

Differential Equations Methods for the Monge–Kantorovich Mass Transfer Problem

We demonstrate that a solution to the classical Monge–Kantorovich problem of optimally rearranging the measure μ+ = f+dx onto μ− = f−dy can be constructed by studying the p-Laplacian equation −div(|Dup| Dup) = f + − f in the limit as p → ∞. The idea is to show up → u, where u satisfies |Du| ≤ 1, −div(aDu) = f − f for some density a ≥ 0, and then to build a flow by solving an ODE involving a, Du...

On Monge - Kantorovich Problem in the Plane

We transfer the celebrating Monge-Kontorovich problem in a bounded domain of Euclidean plane into a Dirichlet boundary problem associated to a quasi-linear elliptic equation with 0−order term missing in its diffusion coefficients: A(x, F ′ x )F ′′ xx +B(y, F ′ y )F ′′ yy = C(x, y, F ′ x , F ′ y ) where A(., .) > 0, B(., .) > 0 and C are functions based on the initial distributions, F is an unkn...

Multivariate Monge-Kantorovich Transportation Problem

Received: January 6, 2011 Accepted: January 24, 2011 doi:10.5539/jmr.v3n2p66 The research is supported by Zhejiang Provincial Education Department research projects (Y201016421) Abstract Monge-Kantorovich transportation problem in a bounded region of Euclidean space is transferred into a partial differential equations group. And then the explicit formula of the optimal coupling is achieved. The...

Minimizing Flows for the Monge-Kantorovich Problem