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 similiar 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).

This event takes place in hybrid form (in person and online on Zoom). Slides and additional materials are available on the Moodle service of the University of Vienna. If you want to participate, please write an email to matteo.tommasini@univie.ac.at