Abstract: Symmetric decreasing rearrangement replaces a given function $f$ on $\mathbb{R}^d$ by a radially decreasing function $f^*$ that is equimeasurable to $f$. Symmetrization techniques have been used to determine the sharp constants in classical functional inequalities such as the Sobolev inequality, and for solving minimization problems in Geometry and Mathematical Physics. Symmetrization also motivates the definition of rearrangement-invariant function spaces.

I will describe recent work with A. Ferone on the extremals of the Polya-Szego inequality. The inequality says that the $p$-norms of the gradient decrease under symmetrization. It is known that there are non-trivial cases of equality, even when $p>1$. We use Ryff's polar factorization to describe these equality cases.

Speaker's homepage: http://www.math.toronto.edu/almut/