## Isoperimetric Problems Between Analysis and Geometry

# Isoperimetric problems with non-local penalization terms

## Berardo Ruffini (Institut Montpelliérain Alexander Grothendieck, Université de Montpellier)

17 jun 2014 -- 14:00

**Abstract.**

The talk will focus on a model arising from the physic problem of describing the behavior of a liquid drop with a charge of $Q > 0$. Such a model takes the form \[ \min\{P (E) + Q^2\mathcal N \mathcal L(E) : vol(E) =\mathrm{constant}\}, \] with $P(E)$ being the perimeter of E ⊂ RN and NL a non-local operator describing the repulsive effect of the charge. We shall discuss existence issues and qualitative behaviors of minimizers depending on the value of Q and on the choice of the non-local operator. We may eventually briefly discuss a similar problem: given a charged closed wire, what shape does it take. The talk is based on collaborative work with M. Goldman and M. Novaga.