On the gradient of Schwarz symmetrization of functions in Sobolev spaces

TL;DR

This paper investigates conditions under which the Schwarz symmetrization of functions in Sobolev spaces remains within the same space, extending classical results using isoperimetric inequalities and analyzing local properties of rearrangements.

Contribution

It provides new sufficient conditions for the preservation of Sobolev space membership under Schwarz symmetrization, generalizing Polya-Szego's theorem.

Findings

Rearranged functions belong locally to the original Sobolev space.

Sufficient conditions are established for the symmetrization to stay in the Sobolev space.

Results are obtained through relative isoperimetric inequalities.

Abstract

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.