Pacific Institute for the Mathematical Sciences - PIMS

A self-dual polar factorization for vector fields

Abstract
We show that any non-degenerate vector field u in L^{\infty}(\Omega,
\R^N), where \Omega is a bounded domain in \R^N, can be written as
{equation} \hbox{u(x)= \nabla_1 H(S(x), x) for a.e. x \in \Omega},
{equation} where S is a measure preserving point transformation on
\Omega such that S^2=I a.e (an involution), and H: \R^N \times \R^N \to
\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 u as u(x)=\nabla \phi
(S(x)), where \phi 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.

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