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

Antunes, Pedro R. S.; Benguria, Rafael D.; Lotoreichik, Vladimir; Ourmieres-Bonafos, Thomas

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

Date

2021

Authors

Antunes, Pedro R. S.

Benguria, Rafael D.

Lotoreichik, Vladimir

Ourmieres-Bonafos, Thomas

Abstract

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