Maximization and minimization of the principal eigenvalue of the Laplacian with indefinite weight under Dirichlet and Robin boundary conditions on classes of rearrangements

Let Ω ⊂ R^N, N ≥ 1, be a bounded connected open set with Lipschitz boundary. We consider the weighted eigenvalue problem − Δu = λmu in Ω with λ ∈ R, m ∈ L^∞(Ω) and with homogeneous Dirichlet and Robin boundary conditions. First, we study weak* continuity, convexity and Gâteaux differentiability of the map m ↦1/λ_1(m), where λ_1(m) is the principal eigenvalue. Then, denoting by G(m_0) the class of rearrangements of a fixed weight m_0 and assuming that m_0 is positive on a set of positive Lebesgue measure, we investigate the minimization and maximization of λ_1(m) over G(m_0). The minimization problem has been already discussed in some papers; here we give an alternative treatment of some known results about the existence and characterization of minimizers of λ_1(m). We underline that our approach allows us to deal with Dirichlet and Robin boundary conditions together. Instead, to our best knowledge, the maximization problem has been only partially addressed in the literature; it turns out that the maximization of λ_1(m) is more intricate than its minimization. In our work we discuss existence, uniqueness and characterization of maximizers both in G(m_0) and in its weak* closure \overline{G(m_0)}. In particular, we provide an original full description of the unique maximizer in the case of Dirichlet boundary conditions. To prove this result we give a generalization of a lemma of Burton [Rearrangements of functions, maximization of convex functionals and vortex rings, Math. Ann. 276 (1987) 225–253, doi:10.1007/bf01450739] about rearrangement of functions, which we did not find in literature. In the context of the population dynamics, this kind of problems arise from the question of determining the optimal spatial location of favorable and unfavorable habitats in order to increase the chances of survival or extinction of a population.