Geodesic
Symmetric decreasing rearrangement is sometimes continuous
Journal of the American Mathematical Society, Tome 02 (1989) no. 4, pp. 683-773

Voir la notice de l'article provenant de la source American Mathematical Society

This paper deals with the operation $\mathcal {R}$ of symmetric decreasing rearrangement which maps ${{\mathbf {W}}^{1,p}}({{\mathbf {R}}^n})$ to ${{\mathbf {W}}^{1,p}}({{\mathbf {R}}^n})$. We show that even though it is norm decreasing, $\mathcal {R}$ is not continuous for $n \geq 2$. The functions at which $\mathcal {R}$ 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 ${{\mathbf {W}}^{1,p}}({{\mathbf {R}}^n})$. Curiously, $\mathcal {R}$ is always continuous in fractional Sobolev spaces ${{\mathbf {W}}^{\alpha ,p}}({{\mathbf {R}}^n})$ with $0 \alpha 1$. 
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Almgren, Frederick J.; Lieb, Elliott H. Symmetric decreasing rearrangement is sometimes continuous. Journal of the American Mathematical Society, Tome 02 (1989) no. 4, pp. 683-773. doi: 10.1090/S0894-0347-1989-1002633-4

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