Shape optimisation of the first solenoidal Maxwell eigenvalue

We study the minimisation of the first solenoidal Maxwell eigenvalue $λ_1^{\mathrm{Max}}(Ω)$ in perfectly conducting cavities through the scale-invariant functional $J_{\mathrm{Per}}(Ω):=\mathrm{Per}(Ω)λ_1^{\mathrm{Max}}(Ω)$ in $\mathbb R^3$. While the corresponding unconstrained problem is known to