Dirac Faber–Krahn inequality with infinite-mass boundary conditions (arbitrary domains and rectangles)

Establish that, for the two-dimensional Dirac operator with infinite-mass boundary conditions, the Faber–Krahn-type inequality holds: among bounded planar domains of fixed area, the disk minimizes the lowest eigenvalue; and, in the class of rectangles of fixed area, the square minimizes the lowest eigenvalue.

Background

The authors reference recent conjectures proposing that the classical Faber–Krahn inequality extends to the relativistic Dirac operator with infinite-mass boundary conditions, both for arbitrary domains and for rectangles. Despite numerical evidence and partial analytical results in cited works, a full proof remains elusive due to the lack of symmetrization tools and the non-separable nature of the rectangular Dirac problem.

This broader context motivates the magnetic Laplacian problem studied here as another challenging geometric spectral optimization question.