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–Ampère 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 \(\mathbb {R}^{d}\) ${\mathbb{R}}^d$ . We exploit the underlying geometrical structure provided by the associated Voronoi decomposition of the space to obtain \(C^{1,1}\) $C^{1,1}$ -regularity for local minimizers out of a finite number of shock times.