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A variational formulation for Dirac operators in bounded domains: applications to spectral geometric inequalities
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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 Szeg\"o 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.
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Springer