Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

 
 

 

Symmetric decreasing rearrangement is sometimes continuous


Authors: Frederick J. Almgren and Elliott H. Lieb
Journal: J. Amer. Math. Soc. 2 (1989), 683-773
MSC: Primary 49F20; Secondary 46E30, 49A50
DOI: https://doi.org/10.1090/S0894-0347-1989-1002633-4
MathSciNet review: 1002633
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: 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 $0 < \alpha < 1$.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC: 49F20, 46E30, 49A50

Retrieve articles in all journals with MSC: 49F20, 46E30, 49A50


Additional Information

Article copyright: © Copyright 1989 American Mathematical Society