Optimizing the ground state energy of the three-dimensional magnetic Dirichlet Laplacian with constant magnetic field

TL;DR

This work shows that in three dimensions the Faber-Krahn-type minimizer for the magnetic Dirichlet Laplacian with a constant field is not a ball; through a combination of analytic results and numerical experiments it demonstrates that minimizers become elongated along the field as the strength $B$ increases, a behavior termed spaghettification. The authors develop a cylindrical-geometry analysis yielding precise asymptotics for optimal cylinder height $h^*(B)$ and energy, and they implement a comprehensive numerical framework based on star-shaped domain parametrization and the Method of Particular Solutions, augmented by axisymmetric simplifications and shape-derivative computations, to identify minimizers and test conjectures. Their results indicate that minimizers are axisymmetric and convex for all $B$, with energy transitions dominated by cylinder-like geometry at large $B$, and they pose open problems on two-term asymptotics for the true minimizers, convergence under rescaling, and sharper degeneration rates. Overall, the paper advances the understanding of spectral shape optimization in magnetic settings and provides practical numerical tools for exploring 3D geometries under strong magnetic fields.

Abstract

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 the minimizers are not expected to 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.

Figures (4)

• Figure 1: $h^*(B)$ and $\lambda_{1,cyl}^*(B)$ as functions of the field strength $B$.
• Figure 2:
• Figure 3: Minimizers $\Omega^*(B)$ for various field strengths. The magnetic field is oriented along the vertical axis (drawn in red).
• Figure 4: Numerically obtained values for the lengths $h(\Omega^*(B))$ and $R(\Omega^*(B))$ compared to the lengths $h^*(B)$ and $R^*(B)=(\pi h^*(B))^{-1/2}$ for optimal cylinders.

Theorems & Definitions (27)

• Proposition 2.1
• proof
• Proposition 2.2
• Definition 2.3
• Remark 1
• Theorem 3.1
• proof
• Lemma 3.2
• proof
• Corollary 3.3