## A variational formulation for Dirac Operators in bounded domains and applications to spectral geometric inequalities

### Abstract

In this talk I will present spectral features of the Dirac operator with infi nite mass boundary conditions in a smooth bounded domain of $\mathbb{R}^2$. A non-linear variational formulation to characterize its principal eigenvalue will be presented. This characterization allows for a simple proof of a Szegö type inequality as well as a new formulation of a Faber-Krahn type inequality for this operator. Moreover, strong numerical evidence supporting the existence of a Faber-Krahn type inequality, will be given.

This talk is based on joint work with Pedro Antunes, Vladimir Lotoreichik, and Thomas Ourmieres-Bonafos.