Fournais–Helffer conjecture on the magnetic Neumann eigenvalue of planar domains

Let $\Omega\subset\mathbb{R}^2$ be a bounded, simply-connected $C^\infty$-smooth domain, let $b>0$ be the intensity of a homogeneous magnetic field, and let $\mu_1^b(\Omega)>0$ denote the lowest eigenvalue of the magnetic Neumann Laplacian on $\Omega$. Let $\mathcal{B}\subset\mathbb{R}^2$ be the disk with the same area as $\Omega$. Fournais–Helffer's conjecture. For every $b>0$, one has

$$\mu_1^b(\Omega)\leq\mu_1^b(\mathcal{B}).$$

This conjecture asserts that among bounded simply-connected planar domains of fixed area, the disk maximizes the lowest magnetic Neumann eigenvalue for a homogeneous magnetic field. It is motivated by the weak- and strong-field asymptotics, while the general optimization problem remains open.