Given a measurable subset , we denote its -dimensional Lebesgue measure by . We will denote by the open ball centered at the origin and having the same measure as , i.e. . The norm of vector will be denoted by . Finally, we will denote by the volume of the unit ball in . It is worth recalling that
where us the usual gamma function.
Definition (Schwarz symmetrization). Let be a bounded domain. Let be a measurable function. Then, its Schwarz symmetrization (or the spherically symmetric and decreasing rearrangement) is the function defined by
Observe that if is the radius of , then
We obviously have the following properties of Schwarz symmetrization (more…)