A Self-dual Polar Factorization for Vector Fields

We show that any non-degenerate vector field u in L∞(Ω,RN), where Ω is a bounded domain in RN , can be written as u(x) = ∇1H(S(x),x) for a.e. x ∈Ω, where S is a measure preserving point transformation on Ω such that S2 = I a.e (an involution), and H : RN ×RN → R is a globally Lipschitz anti-symmetric convex-concave Hamiltonian. Moreover, u is a monotone map if and only if S can be taken to be the identity, which suggests that our result is a self-dual version of Brenier’s polar decomposition for the vector field as u(x) = ∇φ(S(x)), where φ is convex and S is a measure preserving transformation. We also describe how our polar decomposition can be reformulated as a (self-dual) mass transport problem. c © 2000 Wiley Periodicals, Inc.