Optimizing the Ground State Energy of the Three-Dimensional Magnetic Dirichlet Laplacian with Constant Magnetic Field

This paper concerns the shape optimization problem of minimizing the ground state energy of the magnetic Dirichlet Laplacian with constant magnetic field among three-dimensional domains of fixed volume. In contrast to the two-dimensional case, a generalized “magnetic” Faber–Krahn inequality does not hold and minimizers cannot be balls when the magnetic field is turned on. An analysis of the problem among cylindrical domains reveals geometric constraints for general minimizers. In particular, minimizers must elongate with a certain rate along the direction of the magnetic field as the field strength increases. In addition to the theoretical analysis, we present numerical minimizers which confirm this prediction and give rise to further conjectures.