A Self-dual Polar Factorization for Vector Fields

2011/01/26 by Nassif Ghoussoub, Ghoussoub, Nassif, Abbas Moameni +1
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP

paper · pdf · doi:10.48550/arxiv.1101.4979

20 pages, Updated version - if any - can be downloaded at http://www.birs.ca/~nassif/

arxiv created 2011/08/10 · arxiv updated 2011/08/12

Abstract

We show that any non-degenerate vector field u in L^∞(Ω, \R^N), where Ω is a bounded domain in \R^N, can be written as equation \hboxu(x)= ∇₁ H(S(x), x) for a.e. x ∈ Ω, equation where S is a measure preserving point transformation on Ω such that S²=I a.e (an involution), and H: \R^N × \R^N → \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)=∇ 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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