Archiv der Mathematik

, Volume 93, Issue 2, pp 181–190 | Cite as

Explicit approximation of the symmetric rearrangement by polarizations



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.


Symmetric rearrangement Schwarz symmetrization Polarization Two-point rearrangement Pólya–Szegő inequality Approximation of symmetrization Steiner symmetrization Foliated Schwarz symmetrization Spherical cap rearrangement Discrete rearrangement 

Mathematics Subject Classification (2000)

Primary 26D15 Secondary 35A25 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 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. 47–91Google Scholar
  2. 2.
    Brascamp H.J., Lieb E.H., Luttinger J.M.: A general rearrangement inequality for multiple integrals. J. Funct. Anal. 17, 227–237 (1974)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Brock F., Solynin A.Y.: An approach to symmetrization via polarization. Trans. Amer. Math. Soc. 352, 1759–1796 (2000)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Crandall M.G., Tartar L.: Some relations between nonexpansive and order preserving mappings. Proc. Amer. Math. Soc. 78, 385–390 (1980)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    J. A. Crowe, J. A. Zweibel and P. C. Rosenbloom, Rearrangements of functions, J. Funct. Anal. 66 (1986), 432–438.Google Scholar
  6. 6.
    V. N. Dubinin, Transformation of functions and the Dirichlet principle, Mat. Zametki 38 (1985), 49–55, 169.Google Scholar
  7. 7.
    V. N. Dubinin, Transformation of condensers in space, Dokl. Akad. Nauk SSSR 296 (1987), 18–20.Google Scholar
  8. 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.Google Scholar
  9. 9.
    M. A. Krasnosel’skiĭ and J. B. Rutickiĭ, Convex functions and Orlicz spaces, Noordhoff, Groningen, 1961.Google Scholar
  10. 10.
    Pruss A.R.: Discrete convolution-rearrangement inequalities and the Faber-Krahn inequality on regular trees. Duke Math. J. 91, 463–514 (1998)MATHCrossRefMathSciNetGoogle Scholar
  11. 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.Google Scholar
  12. 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.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    A. Y. Solynin, Polarization and functional inequalities, Algebra i Analiz 8 (1996), 148–185.MATHMathSciNetGoogle Scholar
  14. 14.
    G. Talenti, Inequalities in rearrangement invariant function spaces, Nonlinear analysis, function spaces and applications, 5 (Prague, 1994), Prometheus, Prague, 1994, pp. 177–230.Google Scholar
  15. 15.
    Van Schaftingen J.: Universal approximation of symmetrizations by polarizations. Proc. Amer. Math. Soc. 134, 177–186 (2006)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    J. Van Schaftingen, Anisotropic symmetrization, Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (2006), 539–565.MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    J. Van Schaftingen, Approximation of symmetrizations and symmetry of critical points, Topol. Methods Nonlinear Anal. 28 (2006) 61–85.MATHMathSciNetGoogle Scholar
  18. 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.Google Scholar
  19. 19.
    Wolontis V.: Properties of conformal invariants. Amer. J. Math. 74, 587–606 (1952)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  1. 1.Département de MathématiqueUniversité catholique de LouvainLouvain-la-NeuveBelgium

Personalised recommendations