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Polar factorization and monotone rearrangement of vector-valued functions. (English) Zbl 0738.46011
Author’s abstract: Given a probability space \((X,\mu)\) and a bounded domain \(\Omega\) in \(\mathbb{R}^ d\) equipped with the Lebesgue measure \(|\cdot|\) (normalized so that \(|\Omega|=1\)), it is shown (under additional technical assumptions on \(X\) and \(\Omega\)) that for every vector-valued function \(u\in L^ p(X,\mu;\mathbb{R}^ d)\) there is a unique “polar factorization” \(u=\nabla\psi\circ s\), where \(\psi\) is a convex function defined on \(\Omega\) and \(s\) is a measure-preserving mapping from \((X,\mu)\) into \((\Omega,|\cdot|)\), provided that \(u\) is nondegenerate, in the sense that \(\mu(u^{-1}(E))=0\) for each Lebesgue negligible subset \(E\) of \(\mathbb{R}^ d\).
Through this result, the concepts of polar factorization of real matrices, Helmholtz decomposition of vector fields, and nondecreasing rearrangements of vector-valued functions are unified. The Monge-Ampère equation is involved in the polar factorization and the proof relies on the study of an appropriate “Monge-Kantorovich” problem.

MSC:
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46E40 Spaces of vector- and operator-valued functions
28A99 Classical measure theory
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[1] , and , Manifolds, Tensor Analysis, and Applications, Reading, MA, Addison-Wesley, 1983.
[2] , and , On optimization problems with prescribed rearrangements, preprint.
[3] Le problème géometrique de déblais et des remblais, Mémoriale de l’Académie des Sciences Mathématiques, Fascicule XXVII, Paris, 1928.
[4] Mathematical Methods of Game and Economic Theory, North-Holland, Amsterdam, 1979. · Zbl 0452.90093
[5] Isoperimetric Inequalities, Pitman, London, 1980. · Zbl 0454.52005
[6] Brenier, C.R. Acad. Sci. Paris 305 pp 805– (1987)
[7] A combinatorial algorithm for the Euler equation of incompressible flows, 8th International Conf. on Computing Methods, Versailles, 1987.
[8] Opérateurs Maximaux Monotones, North-Holland, Amsterdam, 1973.
[9] Caffarelli, Comm. Pure Appl. Math. 37 pp 369– (1984)
[10] Cullen, J. Atmos. Sci. 41 pp 1477– (1984)
[11] Cullen, QJR Meteorol. Soc. 113 pp 735– (1987)
[12] and , Analyse mathématique et calcul numérique pour les sciences et les techniques, Masson, Paris, 1985.
[13] and , Analyse Convexe et Problèmes Varitionnels, Dunod, Paris. 1974.
[14] , and , Inequalities, Cambridge Univ. Press, 1952.
[15] Kantorovich, Dokl. Akad. Nauk SSSR 37 pp 227– (1942)
[16] Knott, J. Opt. Th. Appl. 43 pp 39– (1984)
[17] Mémoire sur la théorie des déblais et de remblais, Mémoires de l’Académic des Sciences. 1781.
[18] Inégalités Isopérimétriques et Applications en Physique, Hermann, Paris, 1985.
[19] Rachev, Th. Prob. Appl. 49 pp 647– (1985)
[20] Recent results in the theory of probability metrics, to appear.
[21] Real Analysis, MacMillan, New York, 1963.
[22] Ryff, Trans. AMS 117 pp 92– (1965)
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