The Faber–Krahn conjecture for the first positive quantum-dot Dirac eigenvalue

Let $\Omega\subset\mathbb{R}^2$ be a bounded domain with $\mathcal C^2$ boundary. Let $D\subset\mathbb{R}^2$ be a disk with the same area as $\Omega$. For $a>0$, let $\mu_{1,\Omega}(a)$ denote the first min-max level associated with the $\overline\partial$-Robin Laplacian on $\Omega$. The Faber–Krahn conjecture. If $\Omega$ is not a disk, then

$$\mu_{1,D}(a)<\mu_{1,\Omega}(a)$$

for all $a>0$. This conjecture is a reformulation of the open spectral-geometric problem asserting that the disk uniquely minimizes the smallest positive eigenvalue of the planar Dirac operator with infinite mass boundary conditions among domains of prescribed area.