Partial W^2,p regularity for optimal transport maps

2016/06/16 by Shibing Chen, Alessio Figalli, Chen, Shibing +1 · 2 citations
Mathematics · #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.1606.05173

Abstract

We prove that, in the optimal transportation problem with general costs and positive continuous densities, the potential function is always of class W^2,p_loc for any p ≥ 1 outside of a closed singular set of measure zero. We also establish global W^2,p estimates when the cost is a small perturbation of the quadratic cost. The latter result is new even when the cost is exactly the quadratic cost.

Citations

• Boundary regularity of maps with convex potentials

Cited by

• Partial regularity of optimal transport with Coulomb cost
• Boundary regularity of optimal transport maps on convex domains

Related

• Global W^2,p Regularity in Optimal Transport
• Partial regularity for optimal transport with p-cost away from fixed points
• Partial Regularity for optimal transport maps
• Sobolev regularity for optimal transport maps of non-convex planar domains
• Partial regularity of optimal transport with Coulomb cost