Calculus of Variations and Geometric Measure Theory

A. Dumas - F. Santambrogio

Optimal trajectories in $L^1$ and under $L^1$ penalizations

created by santambro on 04 May 2023
modified on 30 Oct 2023


Accepted Paper

Inserted: 4 may 2023
Last Updated: 30 oct 2023

Journal: Comptes Rendus Mathématiques
Year: 2023


Motivated by a MFG model where the trajectories of the agents are piecewise constants and agents pay for the number of jumps, we study a variational problem for curves of measures where the cost includes the length of the curve measures with the $L^1$ distance, as well as other, non-autonomous, cost terms arising from congestion effects. We prove several regularity results (first in time, then in space) on the solution, based on suitable approximation and maximum principle techniques. We then use modern algorithms in non-smooth convex optimization in order to obtain a numerical method to simulate such solutions.