Abstract
We present an elementary proof of an important result of Y. Brenier [Br1, Br2], namely, that vector fields in ℝd 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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Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math. 46, 375–417 (1991).
Y. Brenier, Décomposition polaire et réarrangement monotone des champs de vecteurs, C. R. Acad. Sci. Paris 305, Série I, 805–808, (1987).
B. Dacorogna, Direct Methods in the Calculus of Variations, 1989, Springer-Verlag.
T. Rockafellar, Convex Analysis, 1970, Princeton University Press.
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Communicated by D. Kinderlehrer
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Gangbo, W. An elementary proof of the polar factorization of vector-valued functions. Arch. Rational Mech. Anal. 128, 381–399 (1994). https://doi.org/10.1007/BF00387715
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DOI: https://doi.org/10.1007/BF00387715