A Self‐Dual Polar Factorization for Vector Fields

Abstract

AbstractWe show that any nondegenerate vector field u in \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath, amssymb} \pagestyle{empty} \begin{document} \begin{align*}L^{\infty}(\Omega, \mathbb{R}^N)\end{align*} \end{document}, where Ω is a bounded domain in \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath, amssymb} \pagestyle{empty} \begin{document} \begin{align*}\mathbb{R}^N\end{align*} \end{document}, can be written as \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath, amssymb} \pagestyle{empty} \begin{document} \begin{align*}u(x)= \nabla_1 H(S(x), x)\quad {\text for a.e.\ x \in \Omega}\end{align*} \end{document}}, where S is a measure‐preserving point transformation on Ω such that \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath, amssymb} \pagestyle{empty} \begin{document} \begin{align*}S^2=I\end{align*} \end{document} a.e. (an involution), and \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath, amssymb} \pagestyle{empty} \begin{document} \begin{align*}H: \mathbb{R}^N \times \mathbb{R}^N \to \mathbb{R}\end{align*} \end{document} is a globally Lipschitz antisymmetric 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 \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath, amssymb} \pagestyle{empty} \begin{document} \begin{align*}u(x)=\nabla \phi (S(x))\end{align*} \end{document}, 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. © 2012 Wiley Periodicals, Inc.