Abstract

The circle enjoys several optimality properties: from being the solution to the isoperimetric inequality, to being the shape with the least first frequency, from having the minimum electrostatic capacity, to having the maximal torsional rigidity. Do polygons enjoy similar optimality properties?

In this seminar, we will investigate such a question by considering two prototypical classes of energies. The first, is a nonlocal functional with a Riesz type potential. The second is a crystalline perimeter perturbed with a functional of the former kind. In particular, for the latter we will investigate the case of general kernels in any dimension, while for the former we will focus on the case of Riesz potentials on triangles and quadrilaterals in dimension two.

This talk is based on joint works with Marco Bonacini (Università di Trento) and Ihsan Topaloglu (Virginia Commonwealth University).