Uniqueness of the minimizer for the non-convex magnetic functional

Determine whether the minimization problem μ = inf_{u∈W₀^{1,2}(Ω₁), u≠0} [2‖∂₁^A u‖‖∂₂^A u‖ / ‖u‖²] admits a unique minimizer for a constant magnetic field B and a smooth vector potential A generating B.

Background

The functional J[u] = 2‖∂₁^A u‖‖∂₂^A u‖ is coercive and lower semicontinuous on W₀^{1,2}(Ω₁), ensuring existence of minimizers, but the problem is non-convex and its Euler equation is nonlinear. The authors highlight that proving uniqueness would directly yield the desired symmetry in the symmetric gauge and hence imply the main conjecture about square optimality.

Due to the magnetic operator’s complex-valued nature and lack of positivity-preserving properties, standard techniques for uniqueness fail, leaving this question unresolved.