I recently tried to understand the proof of Faber-Krahn inequality and stumbled upon the Schwarz rearrangement. For example, (See, for example, Henrot's book "Extremum Problems for Eigenvalues of Elliptic Operators", Chapter 2, page 17) There, he define the Schwarz rearrangement $u^*$ of a nonnegative measurable function $u$ on a measurable set $\Omega$ and vanishing on the boundary $\partial\Omega$. It can be shown that
1) The $L^p$ norm remains unchanged, i.e. $\|u\|_p = \|u^*\|_p$.
2) We have the Polya inequality, which states that if $u\in W^{1,p}(\Omega)$, then $u^*\in W^{1,p}(\Omega)$ and $$ \|\nabla u^*\|_{p,\Omega^*}\le \|\nabla u\|_{p,\Omega} $$ where $\Omega^*$ is the Schwarz rearrangement of $\Omega$ and $W^{1,p}(\Omega)$ the Sobolev space.
Two questions:
1) Can we still define Schwarz rearrangement of any measurable function, without the nonnegativity assumption?
2) If the answer to Q1 is positive, then does it mean that the two results I quote above fail to hold without the nonnegativity assumption? If so, can someone provide a counter-example?