An elementary proof of the polar factorization of vector-valued functions
نویسنده
چکیده
We present an elementary proof of an important result of Y. Brenier ([Br1], [Br2]), namely that vector fields in R satisfying a nondegeneracy condition admit the polar factorization (∗) u(x) = ∇ψ(s(x)), where ψ is a convex function and s is a measure-preserving mapping. Brenier solves a minimization problem using Monge-Kantorovich theory, whereas we turn our attention to a dual problem, whose Euler-Lagrange equation turns out to be (∗).
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