On Shape Optimization with Large Magnetic Fields in Two Dimensions

Area of Science:

• Mathematical Physics
• Spectral Theory
• Differential Geometry

Background:

• Eigenvalues of magnetic Laplacians are crucial in understanding quantum systems and wave phenomena.
• Domain geometry significantly influences spectral properties, particularly under external fields.
• Symmetry principles often emerge in optimal configurations within physical systems.

Purpose of the Study:

• To investigate the relationship between domain shape and magnetic eigenvalues in the strong magnetic field limit.
• To establish asymptotic bounds for magnetic eigenvalues and their connection to domain asymmetry.
• To demonstrate that optimal domains for magnetic Laplacians tend towards symmetric shapes.

Main Methods:

• Derivation of several asymptotic bounds on magnetic eigenvalues.
• Analysis of the magnetic Dirichlet Laplacian for planar domains and rectangles.
• Investigation of the magnetic Dirac operator with infinite mass boundary conditions.
• Estimation of the torsion function on rectangles.

Main Results:

• Established that for a bounded simply-connected planar domain, if the n-th eigenvalue of the magnetic Dirichlet Laplacian is less than that of a disk of equal area, its Fraenkel asymmetry approaches zero in the strong magnetic field limit.
• Extended comparable results to rectangles and smooth domains with the magnetic Dirac operator.
• Provided a novel estimate for the torsion function on rectangular domains.

Conclusions:

• The study confirms that in the presence of strong magnetic fields, domains that optimize magnetic eigenvalues tend to exhibit increased symmetry.
• Asymptotic eigenvalue analysis provides a powerful tool for understanding geometric spectral properties.
• The findings have implications for spectral optimization problems in mathematical physics.