arXiv subjects
Publications and source records attributed to Arghya Rakshit.
We construct solutions to Monge-Ampère equations whose Monge-Ampère measures contain singular components supported on low codimensional sets. We also study the regularity of such solutions. To motivate our construction, we present examples arising from optimal transport where the potential of optimal transport maps satisfy Monge-Ampère equations similar to the ones we study.
We prove a pointwise $C^{2,\,α}$ estimate for the potential of the optimal transport map in the case that the densities are only close to constant in a certain $L^p$ sense.
We prove a sharp global $W^{2,\,p}$ estimate for potentials of optimal transport maps that take a certain class of non-convex planar domains to convex ones.
We construct new examples of Monge-Ampère metrics with polyhedral singular structures, motivated by problems related to the optimal transport of point masses and to mirror symmetry. We also analyze the stability of the singular structures under small perturbations of the data given in the problem under consideration.