A variational formulation for Dirac operators in bounded domains. Applications to spectral geometric inequalities

Abstract

We investigate spectral features of the Dirac operator with infinite mass boundary conditions in a smooth bounded domain of ${\mathbb{R}}^2$. 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 Szegö 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.

34 pages, 4 figures

References in corpus (3)

• On the eigenvalues of operators with gaps. Application to Dirac operators
• Two-dimensional Dirac operators with singular interactions supported on closed curves
• The Hijazi inequality on manifolds with boundary

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