The (Non) Continuity of Symmetric Decreasing Rearrangement

Frederick J. Almgren, Elliott Lieb

Research output: Contribution to journalArticlepeer-review


The operation R of symmetric decreasing rearrangement maps W1,p(ℝn) to W1,p(ℝn). 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 W1,p(Rn).

Original languageEnglish (US)
Pages (from-to)183-200
Number of pages18
JournalMathematics in Science and Engineering
Issue numberC
StatePublished - Jan 1 1992

All Science Journal Classification (ASJC) codes

  • General Mathematics
  • General Engineering


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