Isoperimetric inequalities for the lowest magnetic Steklov eigenvalue

This paper studies the optimization of the lowest eigenvalue of the magnetic Steklov problem on planar domains. In the bounded domain setting and for magnetic fields of moderate strengths, we prove that among all simply-connected smooth domains of given area, the disk maximises the lowest magnetic Steklov eigenvalue. For exterior domains, we establish a similar isoperimetric inequality for magnetic fields of moderate strength under fixed perimeter constraint and additional geometric and symmetry assumptions. The proofs rely on the method of torsion-type trial functions in the bounded domain case and on the method of trial functions dependent only on the distance to the boundary in the exterior domain case.

💡 Research Summary

The paper investigates an isoperimetric problem for the lowest eigenvalue of the magnetic Steklov operator on planar domains. In the classical (non‑magnetic) setting, the Steklov eigenvalues arise from the Laplace equation Δu = 0 together with the boundary condition ∂_ν u = σu, where σ is the Steklov eigenvalue. When a constant magnetic field B is present, the differential operator is replaced by the magnetic Laplacian Δ_A = (∇ + iA)·(∇ + iA) with A a vector potential satisfying curl A = B, and the boundary condition becomes ∂_ν_A u = σu. The authors consider two geometric configurations: (i) bounded, simply‑connected smooth domains Ω of prescribed area, and (ii) exterior domains Ω^c = ℝ² \ Ω̅ of prescribed perimeter. Their main results show that, provided the magnetic field strength is “moderate’’ (i.e., below a domain‑dependent threshold), the disk maximises the lowest magnetic Steklov eigenvalue among all admissible shapes in both settings.

Bounded domains.
Let Ω ⊂ ℝ² be a smooth, simply‑connected region with |Ω| = πR². Theorem 1.1 states that if |B| < B₀(R) – a constant that depends on the diameter and curvature of Ω – then
 λ₁(A,Ω) ≤ λ₁(A,D_R),
where D_R denotes the disk of radius R. The proof uses a torsion‑type trial function τ(x) = (|x|² − R²)/4, modified by a complex phase e^{iθ(x)} that compensates for the magnetic potential (∇θ = A). The Rayleigh quotient
 R