Calculus of Variations and Geometric Measure Theory

L. Ambrosio - F. Vitillaro - D. Trevisan

Sharp PDE estimates for random two-dimensional bipartite matching with power cost function

created by trevisan on 05 Jul 2024



Inserted: 5 jul 2024

Year: 2024

ArXiv: 2405.09397 PDF


We investigate the random bipartite optimal matching problem on a flat torus in two-dimensions, considering general strictly convex power costs of the distance. We extend the successful ansatz first introduced by Caracciolo et al. for the quadratic case, involving a linear Poisson equation, to a non-linear equation of $q$-Poisson type, allowing for a more comprehensive analysis of the optimal transport cost. Our results establish new asymptotic connections between the energy of the solution to the PDE and the optimal transport cost, providing insights on their asymptotic behavior.