A mixed formulation of the Monge-Kantorovich equations.(English)Zbl 1132.35333

Authors’ summary: We introduce and analyse a mixed formulation of the Monge-Kantorovich equations, which express optimality conditions for the mass transportation problem with cost proportional to distance. Furthermore, we introduce and analyse the finite element approximation of this formulation using the lowest order Raviart-Thomas element. Finally, we present some numerical experiments, where both the optimal transport density and the associated Kantorovich potential are computed for a coupling problem and problems involving obstacles and regions of cheap transportation.

MSC:

 35D05 Existence of generalized solutions of PDE (MSC2000) 35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000) 49J40 Variational inequalities 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 82B27 Critical phenomena in equilibrium statistical mechanics

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