Lack of regularity of the transport density in the monge problem

2017/04/24 by Dweik, Samer
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1704.07180

Abstract

In this paper, we provide a family of counterexamples to the regularity of the transport density in the classical Monge-Kantorovich problem. We prove that the W^1,p regularity of the source and target measures f ^± does not imply that the transport density σ is W^1,p , that the BV regularity of f ^± does not imply that σ is BV and that f^± ∈ C^∞ does not imply that σ is W^1,p , for large p.

Related

• L^p,q estimates on the transport density
• Obstructions to regularity in the classical Monge problem
• A General Duality Theorem for the Monge--Kantorovich Transport Problem
• Backward Monge Potential and Monge-Ampere Equation
• Intrinsic Sparsity of Kantorovich Solutions