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).
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© 1990 Springer Science+Business Media New York
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Almgren, F.J., Lieb, E.H. (1990). The (Non)Continuity of Symmetric Decreasing Rearrangement. In: Berestycki, H., Coron, JM., Ekeland, I. (eds) Variational Methods. Progress in Nonlinear Differential Equations and Their Applications, vol 4. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4757-1080-9_1
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DOI: https://doi.org/10.1007/978-1-4757-1080-9_1
Publisher Name: Birkhäuser, Boston, MA
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