The (Non) Continuity of Symmetric Decreasing Rearrangement

Frederick J. Almgren; Elliott H. Lieb

doi:10.1016/S0076-5392(08)63380-9

The (Non) Continuity of Symmetric Decreasing Rearrangement

Frederick J. Almgren
, Elliott H. Lieb

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

The operation R of symmetric decreasing rearrangement maps W^1,p(ℝⁿ) to W^1,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 W^1,p(Rⁿ).

Original language	English (US)
Pages (from-to)	183-200
Number of pages	18
Journal	Mathematics in Science and Engineering
Volume	186
Issue number	C
DOIs	https://doi.org/10.1016/S0076-5392(08)63380-9
State	Published - Jan 1 1992

All Science Journal Classification (ASJC) codes

General Mathematics
General Engineering

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abstract = "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).",

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The (Non) Continuity of Symmetric Decreasing Rearrangement

Abstract

All Science Journal Classification (ASJC) codes

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