Partial regularity for minimizers of a class of discontinuous Lagrangians

We study a one dimensional Lagrangian problem including the variational reformulation, derived in a recent work of Ambrosio-Baradat-Brenier, of the discrete Monge-Amp\`ere gravitational model, which describes the motion of interacting particles whose dynamics is ruled by the optimal transport problem. The more general action type functional we consider contains a discontinuous potential term related to the descending slope of the opposite squared distance function from a generic discrete set in R^{d}. We exploit the underlying geometrical structure provided by the associated Voronoi decomposition of the space to obtain C^{1,1} regularity for local minimizers out of a finite number of shock times.