Minimising convex combinations of low eigenvalues∗. (7th April 2014) - Iversen, Mette; Mazzoleni, Dario
Abstract : We consider the variational problem         inf{ αλ 1 ( Ω ) +  βλ 2 ( Ω ) + (1 −  α  −  β ) λ 3 ( Ω ) | Ω open in ℝ n,  | Ω | ≤ 1}, for α, β  ∈ [0, 1], α  +  β  ≤ 1, where λ k ( Ω ) is the k th eigenvalue of the Dirichlet Laplacian acting in L 2 ( Ω ) and | Ω | is the Lebesgue measure of Ω . We investigate for which values of α, β every minimiser is connected.