Nicola Fusco, Dip. Mat. Univ. Napoli

"Polya-Szego type inequalities for Steiner symmetrization: cases of equality"

The classical Polya-Szego inequality says that if is a nonnegative smooth function with compact support and is its Steiner symmetral, then

$\begin{displaymath} \int\vert\nabla u\vert^p\,dx\geq\int\vert\nabla u^s\vert^p\,dx \end{displaymath}$

for all

. In particular, when

is the characteristic function of a set

of finite perimeter, the previous inequality implies that $P(E)\geq P(E^s) \,,$ where

denotes the perimeter of

and

is the Steiner symmetral of

with respect to a given hyperplane.

We shall discuss what can be said about the case of equality holds in the two previous inequalities.

TUESDAY, February 24, 2004
Time: 1:30 P.M.
Location: PPB 300