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)


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 ).


Sobolev Inequality Isoperimetric Inequality Singular Part Clockwise Order Fractional Sobolev Space 
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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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