We consider the variational problem
inf{αλ1(Ω) + βλ2(Ω) + (1 − α − β)λ3(Ω) | Ω open in ℝn, |Ω| ≤ 1},
for α, β ∈ [0, 1], α + β ≤ 1, where
λk(Ω)
is the kth
eigenvalue of the Dirichlet Laplacian acting in L2(Ω) and
|Ω| is
the Lebesgue measure of Ω. We investigate for which values of α, β every minimiser is
connected.