A Self‐Dual Polar Factorization for Vector Fields |
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Authors: | Nassif Ghoussoub Abbas Moameni |
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Institution: | 1. University of British Columbia, Department of Mathematics, Vancouver BC V6T 1Z2, CANADA;2. University of Lethbridge, Department of Mathematics and Computer Science, Lethbridge AB T1K 3M4, CANADA |
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Abstract: | We show that any nondegenerate vector field u in \begin{align*}L^{\infty}(\Omega, \mathbb{R}^N)\end{align*} , where Ω is a bounded domain in \begin{align*}\mathbb{R}^N\end{align*} , can be written as \begin{align*}u(x)= \nabla_1 H(S(x), x)\quad {\text for a.e.\ x \in \Omega}\end{align*} }, where S is a measure‐preserving point transformation on Ω such that \begin{align*}S^2=I\end{align*} a.e. (an involution), and \begin{align*}H: \mathbb{R}^N \times \mathbb{R}^N \to \mathbb{R}\end{align*} 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 \begin{align*}u(x)=\nabla \phi (S(x))\end{align*} , where ? 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. |
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