Symmetry breaking in the minimization of the first eigenvalue for the composite clamped punctured disk

Let D0={x∈R2:0<|x|<1} be the unit punctured disk. We consider the first eigenvalue λ1(ρ) of the problem Δ2u=λρu in D0 with Dirichlet boundary condition, where ρ is an arbitrary function that takes only two given values 0<α<β and is subject to the constraint ∫D0ρdx=αγ+β(|D0|−γ) for a fixed 0<γ<|D0|. We will be concerned with the minimization problem ρ↦λ1(ρ). We show that, under suitable conditions on α, β and γ, the minimizer does not inherit the radial symmetry of the domain.