A Self-dual Polar Factorization for Vector Fields

We show that any non-degenerate vector field $u$ in $L^{\infty}(Ω, \R^N)$, where $Ω$ is a bounded domain in $\R^N$, can be written as {equation} \hbox{$u(x)= \nabla_1 H(S(x), x)$ for a.e. $x \in Ω$}, {equation} where $S$ is a measure preserving point transformation on $Ω$ 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 $u(x)=\nabla ϕ(S(x))$, 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.

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