Abstract
Given two probability measures μ and ν we consider a mass transportation mapping T satisfying 1) T sends μ to ν, 2) T has the form T=φ∇φ|∇φ|, where φ is a function with convex sublevel sets. We prove a change of variables formula for T. We also establish Sobolev estimates for φ, and a new form of the parabolic maximum principle. In addition, we discuss relations to the Monge–Kantorovich problem, curvature flows theory, and parabolic non-linear PDE's.
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