Filippo Cagnetti (University of Sussex)

Rigidity of the perimeter and Pólya-Szegő inequalities under circular rearrangement

Abstract:

We discuss circular symmetrisation of sets and the corresponding rearrangement for Sobolev functions. For sets, we first show that circular symmetrisation does not increase the perimeter. After that, we give necessary and sufficient conditions for rigidity. In this context, we say that rigidity holds if the only sets whose perimeter is preserved are symmetric. For Sobolev functions, we show that circular rearrangement does not increase the L^p norm of the gradient. Our analysis can be applied to functionals whose integrand is a convex function of the gradient, and to functions that do not necessarily satisfy Dirichlet boundary conditions. This generalises previous contributions by Pólya and several authors. After that, we study rigidity of the inequality. That is, we discuss under which conditions all the extremals of the inequality are symmetric. This is work in collaboration with Georgios Domazakis (University of Sussex), Matteo Perugini (University of Milan), and Francis Seuffert (University of Pennsylvania).