Abstract
AbstractWe 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 $$\mathbb {R}^{d}$$
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.
Publisher
Springer Science and Business Media LLC
Reference15 articles.
1. Ambrosio, L., Ascenzi, O., Buttazzo, G.: Lipschitz regularity for minimizers of integral functionals with highly discontinuous integrands. J. Math. Anal. Appl. 142, 301–316 (1989)
2. Ambrosio, L., Baradat, A., Brenier, Y.: Monge–Ampère gravitation as a $$\Gamma $$-limit of good rate functions. Anal. PDE 9, 2005–2040 (2023)
3. Ambrosio, L., Baradat, A., Brenier, Y.: $$\Gamma $$-convergence for a class of action functionals induced by gradients of convex functions. Rend. Lincei. Mat. Appl. 32, 97–108 (2021)
4. Ambrosio, L., Brena, C.: Stability of a class of action functionals depending on convex functions. Discrete Contin. Dyn. Syst. 43, 993–1005 (2023)
5. Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edn. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel (2008)