On the gradient of Schwarz symmetrization of functions in Sobolev spaces

Details On the gradient of Schwarz symmetrization of functions in Sobolev spaces On the gradient of Schwarz symmetrization of functions in Sobolev spaces

2010-02-15

arxiv.org/abs/1002.2856v1

Boll. Un. Mat. Ital. B (7) 7 (1993), no. 2, 413-430

Mathematics

Analysis of PDEs

11 pages

Scientific paper

Let S be a Sobolev or Orlicz-Sobolev space of functions not necessarily vanishing at the boundary of the domain. We give sufficient conditions on a nonnegative function in S in order that its spherical rearrangement ("Schwartz symmetrization") still belongs to S. These results are obtained via relative isoperimetric inequalities and somewhat generalize a well-known Polya-Szego's theorem. We also prove that the rearrangement of any function in S is locally in S.

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