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Explicit approximation of the symmetric rearrangement by polarizations

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Abstract

We give an explicit sequence of polarizations such that for every measurable function, the sequence of iterated polarizations converge to the symmetric rearrangement of the initial function.

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References

  1. A. Baernstein, II, A unified approach to symmetrization, Partial differential equations of elliptic type (Cortona, 1992), Sympos. Math., XXXV, Cambridge University Press, Cambridge, 1994, pp. 47–91

  2. Brascamp H.J., Lieb E.H., Luttinger J.M.: A general rearrangement inequality for multiple integrals. J. Funct. Anal. 17, 227–237 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  3. Brock F., Solynin A.Y.: An approach to symmetrization via polarization. Trans. Amer. Math. Soc. 352, 1759–1796 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  4. Crandall M.G., Tartar L.: Some relations between nonexpansive and order preserving mappings. Proc. Amer. Math. Soc. 78, 385–390 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  5. J. A. Crowe, J. A. Zweibel and P. C. Rosenbloom, Rearrangements of functions, J. Funct. Anal. 66 (1986), 432–438.

    Google Scholar 

  6. V. N. Dubinin, Transformation of functions and the Dirichlet principle, Mat. Zametki 38 (1985), 49–55, 169.

    Google Scholar 

  7. V. N. Dubinin, Transformation of condensers in space, Dokl. Akad. Nauk SSSR 296 (1987), 18–20.

    Google Scholar 

  8. V. N. Dubinin, Transformations of condensers in an n-dimensional space, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 196 (1991), Modul. Funktsii Kvadrat. Formy. 2, 41–60, 173.

  9. M. A. Krasnosel’skiĭ and J. B. Rutickiĭ, Convex functions and Orlicz spaces, Noordhoff, Groningen, 1961.

  10. Pruss A.R.: Discrete convolution-rearrangement inequalities and the Faber-Krahn inequality on regular trees. Duke Math. J. 91, 463–514 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  11. M. M. Rao and Z. D. Ren, Theory of Orlicz spaces, Monographs and Textbooks in Pure and Applied Mathematics, 146, Marcel Dekker, New York, 1991.

  12. D. Smets and M. Willem, Partial symmetry and asymptotic behavior for some elliptic variational problems, Calc. Var. Partial Differential Equations 18 (2003), 57–75.

    Article  MATH  MathSciNet  Google Scholar 

  13. A. Y. Solynin, Polarization and functional inequalities, Algebra i Analiz 8 (1996), 148–185.

    MATH  MathSciNet  Google Scholar 

  14. G. Talenti, Inequalities in rearrangement invariant function spaces, Nonlinear analysis, function spaces and applications, 5 (Prague, 1994), Prometheus, Prague, 1994, pp. 177–230.

  15. Van Schaftingen J.: Universal approximation of symmetrizations by polarizations. Proc. Amer. Math. Soc. 134, 177–186 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  16. J. Van Schaftingen, Anisotropic symmetrization, Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (2006), 539–565.

    Article  MATH  MathSciNet  Google Scholar 

  17. J. Van Schaftingen, Approximation of symmetrizations and symmetry of critical points, Topol. Methods Nonlinear Anal. 28 (2006) 61–85.

    MATH  MathSciNet  Google Scholar 

  18. J. Van Schaftingen and M. Willem, Set transformations, symmetrizations and isoperimetric inequalities, Nonlinear analysis and applications to physical sciences, Springer Italia, Milan, 2004, pp. 135–152.

  19. Wolontis V.: Properties of conformal invariants. Amer. J. Math. 74, 587–606 (1952)

    Article  MATH  MathSciNet  Google Scholar 

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  1. Département de Mathématique, Université catholique de Louvain, Chemin du Cyclotron, 2, 1348, Louvain-la-Neuve, Belgium

    Jean Van Schaftingen

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  1. Jean Van Schaftingen
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Correspondence to Jean Van Schaftingen.

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Van Schaftingen, J. Explicit approximation of the symmetric rearrangement by polarizations. Arch. Math. 93, 181–190 (2009). https://doi.org/10.1007/s00013-009-0018-3

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  • DOI: https://doi.org/10.1007/s00013-009-0018-3

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