Some remarks on the optimization of eigenvalue problems involving the p-Laplacian
Wacław Pielichowski · 2008
Abstract. Given a bounded domain Ω ⊂ Rn, numbers p> 1, α ≥ 0 and A ∈ [0, |Ω|], consider the optimization problem: find a subset D ⊂ Ω, of measure A, for which the first eigenvalue of the operator u 7 → −div(|∇u|p−2∇u)+αχD|u|p−2u with the Dirichlet boundary condition is as small as possible. We show that the optimal configuration D is connected with the corresponding positive eigenfunction u in such a way that there exists a number t ≥ 0 for which D = {u ≤ t}. We also give a new proof of symmetry of optimal solutions in the case when Ω is Steiner symmetric and p = 2.