Calculus of Variations and Geometric Measure Theory

L. Portinale - F. Quattrocchi

Discrete-to-continuum limits of optimal transport with linear growth on periodic graphs

created by quattrocchi on 30 Nov 2023



Inserted: 30 nov 2023

Year: 2023

ArXiv: 2311.17284 PDF


We prove discrete-to-continuum convergence for dynamical optimal transport on $\mathbb{Z}^d$-periodic graphs with energy density having linear growth at infinity. This result provides an answer to a problem left open by Gladbach, Kopfer, Maas, and Portinale (Calc Var Partial Differential Equations 62(5), 2023), where the convergence behaviour of discrete boundary-value dynamical transport problems is proved under the stronger assumption of superlinear growth. Our result extends the known literature to some important classes of examples, such as scaling limits of 1-Wasserstein transport problems. Similarly to what happens in the quadratic case, the geometry of the graph plays a crucial role in the structure of the limit cost function, as we discuss in the final part of this work, which includes some visual representations.