A Variational Formulation for Dirac Operators in Bounded Domains. Applications to Spectral Geometric Inequalities

We investigate spectral features of the Dirac operator with infinite mass boundary conditions in a smooth bounded domain $\mathbb{R}^2$. Motivated by geometric inequalities, we prove non-linear variational formulation to characterize its principal eigenvalue. This characterization turns out be very robust and allows for simple proof Szeg\"o type inequality as well new reformulation Faber-Krahn this operator. The paper is complemented strong numerical evidences supporting existence inequality.