• DOI:10.1515/crll.1988.384.153
• Corpus ID: 117568079

Minimal rearrangements of Sobolev functions.

@article{Ziemer1987MinimalRO,
  title={Minimal rearrangements of Sobolev functions.},
  author={William P. Ziemer and John Brothers},
  journal={Journal f{\"u}r die reine und angewandte Mathematik (Crelles Journal)},
  year={1987},
  volume={1988},
  pages={153 - 179}
}

• W. Ziemer, J. Brothers
• Published 1987
• Mathematics
• Journal für die reine und angewandte Mathematik (Crelles Journal)

where α (n) is the volume of the unit η-ball in H$ and μ(ί)< oo is the Lebesgue measure of the set Et = {x: u(x)>t}. Note that μ(ι) = \Ε*\ where E* = {x: ii*(x)>f} and |E*| denotes the Lebesgue measure of E*. The purpose of this paper is to show that if μ is absolutely continuous and equality holds in (1), then u is almost every where equal to a translate of u*. We also construct a C°° example which shows that this may not be true if μ is continuous but not absolutely continuous. More generally, we establish the following result. For l ^ p < oo, let A: [0, oo) — * [0, oo) be of class C and such that