A connection between quantum dot Dirac operators and $\overline\partial$-Robin Laplacians in the context of shape optimization problems

TL;DR

This paper explores Faber-Krahn inequalities for quantum dot Dirac operators and their equivalence to inequalities for arle-Robin Laplacians, establishing results for simply connected domains and boundary conditions.

Contribution

It establishes a novel connection between quantum dot Dirac operators and arle-Robin Laplacians, proving new inequalities for specific domain classes.

Findings

Proved Faber-Krahn inequalities for quantum dot Dirac operators on simply connected domains.

Demonstrated the equivalence between inequalities for Dirac operators and arle-Robin Laplacians.

Analyzed the case of negative mass in the context of these inequalities.

Abstract

This work addresses Faber-Krahn-type inequalities for quantum dot Dirac operators with nonnegative mass on bounded domains in $\mathbb{R}^2$. We show that this family of inequalities is equivalent to a family of Faber-Krahn-type inequalities for $\overline\partial$-Robin Laplacians. Thanks to this, we prove them in the case of simply connected domains for quantum dot boundary conditions asymptotically close to zigzag boundary conditions. Finally, we also study the case of negative mass.