The (Non)Continuity of Symmetric Decreasing Rearrangement

  • Frederick J. AlmgrenJr.
  • Elliott H. Lieb
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 4)

Abstract

The operation R of symmetric decreasing rearrangement maps W 1,p (R n ) to W 1,p (R n ). Even though it is norm decreasing we show that R is not continuous for n ⩾ 2. The functions at which R is continuous are precisely characterized by a new property called co-area regularity. Every sufficiently differentiable function is co-area regular, and both the regular and the irregular functions are dense in W 1,p (R n ).

Keywords

Sobolev Inequality Isoperimetric Inequality Singular Part Clockwise Order Fractional Sobolev Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [AL]
    F. Almgren and E. LiebSymmetric decreasing rearrangement is sometimes continuous, in preparation.Google Scholar
  2. [B]
    C. Bandle, Isoperimetric inequalities and applications, Pitman (Boston, London, Melbourne) 1980.Google Scholar
  3. [BZ]
    J. Brothers and W. Ziemer, Minimal rearrangements of Sobolev functions, Jour. Reine Angew. Math. 384, 153–179 (1988).MathSciNetMATHGoogle Scholar
  4. [CG]
    G. Chiti, Rearrangements of functions and convergence in Orlicz spaces, Appl. Anal. 9, 23–27 (1979).MathSciNetMATHGoogle Scholar
  5. [CJ]
    J-M. Coron, The continuity of the rearrangement in W 1,P(R), Ann. Scuol. Norm. Sup. Pisa, Ser 4, 11, 57–85 (1984).Google Scholar
  6. [H]
    K. Hilden, Symmetrization of functions in Sobolev spaces and the isoperimetric inequality, Manuscr. Math. 18, 215–235 (1976).Google Scholar
  7. [K]
    B. Kawohl, Rearrangements and convexity of level sets in partial differential equations, Lect. Notes in Math. 1150, Springer (Berlin, Heidelberg, New York ) 1985.Google Scholar
  8. [L]
    E. Lieb, Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation,Stud. Appl. Math. 57 93–105 (1977). See appendix.Google Scholar
  9. [M]
    J. Michael, Lipschitz approximations to summable functions, Acta Math. 111, 73–94 (1964).MathSciNetCrossRefMATHGoogle Scholar
  10. [PS]
    G. Polya and G. Szegö, Isoperimetric inequalities in mathematical physics, Ann. Math. Stud. 27, Princeton University Press (Princeton) (1951).Google Scholar
  11. [SJ]
    J. Serrin, On the definition and properties of certain variational integrals, Trans. Amer. Math. Soc. 101, 139–167, (1961).MathSciNetCrossRefGoogle Scholar
  12. [S1]
    E. Sperner, Zur symmetrisierung von Funktionen auf Sphären, Math. Z. 134, 317–327 (1973).MathSciNetCrossRefGoogle Scholar
  13. [S2]
    E. Sperner, Symmetrisierung für Funktionen mehrerer reeller Variablen, Manuscr. Math. 11, 159–170 (1974).Google Scholar
  14. [T]
    G. Talenti, Best constant in Sobolev inequality, Ann. Pura Appl. 110, 353–372 (1976).CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1990

Authors and Affiliations

  • Frederick J. AlmgrenJr.
    • 1
  • Elliott H. Lieb
    • 1
  1. 1.Department of MathematicsPrinceton UniversityPrincetonUSA

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