Symmetry of minimizers for the non-convex magnetic functional in the symmetric gauge

Establish, in the symmetric gauge A(x) = (−(1/2) x₂, (1/2) x₁) B (i.e., θ = 1/2), the existence of a minimizer u ∈ W₀^{1,2}(Ω₁) for μ = inf_{u≠0} [2‖∂₁^A u‖‖∂₂^A u‖ / ‖u‖²] that satisfies the symmetry ‖∂₁^A u‖ = ‖∂₂^A u‖.

Background

To connect spectral optimality to symmetry, the authors lower-bound the principal eigenvalue by the product of magnetic gradient norms, yielding a non-convex minimization problem for μ. They conjecture that, in the symmetric gauge, a minimizer can be chosen to have equal magnetic gradient norms in the two coordinate directions, which, if true, would imply the square’s optimality among rectangles via a variational comparison.

They note the minimizer exists (by coercivity and lower semicontinuity) and satisfies an associated nonlinear Euler–Lagrange-type equation, but uniqueness and the conjectured symmetry remain unresolved.

References

If θ=\frac{1}{2}, then there exists a minimiser~u of the right-hand side of~area which satisfies \n\begin{equation}\label{symmetry} |\partial_1^A u| = |\partial_2^A u| \,. \end{equation} Because of the non-linear structure of the minimisation problem and the lack of positivity preserving property for the magnetic Laplacian, we have been able to establish neither the uniqueness of the minimiser nor Conjecture~\ref{Conj.symmetry}, respectively.

symmetry:

$\|\partial_1^A u\| = \|\partial_2^A u\| \,.$

area:

$\lambda_1^B(\Omega_a) \geq \inf_{\stackrel[u \not= 0]{}{u \in W_0^{1,2}(\Omega_1) }} \frac{2 \, \|\partial_1^A u\| \, \|\partial_2^A u\|}{\, \|u\|^2} =: \mu \,,$

— Is the optimal magnetic rectangle a square? (2508.16152 - Krejcirik, 22 Aug 2025) in Section 3 (From symmetry to optimality), Conjecture 2

Symmetry of minimizers for the non-convex magnetic functional in the symmetric gauge

Background

References

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