In this paper, we investigate spectral isoperimetric inequalities for the Robin Laplacian with or without a magnetic field in smooth, simply connected planar domains. By revisiting known results and introducing new geometric bounds, we explore the interplay between the geometry of the domain, the magnetic field intensity, and the Robin parameter. Specifically, we analyze the behavior of the lowest eigenvalue under area or perimeter constraints, highlighting cases where the disk ceases to be the optimizer. Through a combination of theoretical proofs and numerical computations, we demonstrate oscillations in the spectral isoperimetric inequality as the magnetic field varies, particularly for ellipses. Additionally, we formulate new geometric bounds for the Robin Laplacian without a magnetic field and extend quantitative reverse Faber–Krahn inequalities to ellipses with Neumann boundary conditions.
Najem et al. (Thu,) studied this question.