The disk minimisation conjecture for the lowest positive Dirac eigenvalue

Let $m\geq 0$, let $\Omega\subset\mathbb{R}^2$ be an open connected set with locally Lipschitz boundary, and let $\lambda_1(\Omega)$ denote the lowest positive eigenvalue of the Dirac operator with infinite-mass boundary conditions. For a disk $\Omega^*$, impose either equality of area or equality of perimeter with $\Omega$. Disk minimisation conjecture. For every $m\geq 0$,

$$\lambda_1(\Omega)\geq\lambda_1(\Omega^*)$$

when $\Omega^*$ has the same area as $\Omega$, and

$$\lambda_1(\Omega)\geq\lambda_1(\Omega^*)$$

when $\Omega^*$ has the same perimeter as $\Omega$. This is the relativistic analogue of the Faber–Krahn inequality. The area-constrained massless case was previously stated as a conjecture, and the general problem remains open; it has been identified as a hot open problem in spectral geometry.