The optimization of eigenvalue problems involving the -Laplacian.

Given a bounded domain Ω ⊂ R and numbers p > 1, α ≥ 0, A ∈ [0, |Ω|], consider the following optimization problem: find a subset D ⊂ Ω, of measure A, for which the first eigenvalue of the operator −∆p + α χD φp with the Dirichlet boundary condition is as small as possible. We prove the existence of optimal solutions and study their qualitative properties. We also obtain the radial symmetry of optimal solutions in the case when Ω is a ball.