Maximization and minimization of the principal eigenvalue of the Laplacian with indefinite weight under Dirichlet and Robin boundary conditions on classes of rearrangements
Let $\Omega\subset\mathbb{R}^N$, $N\geq 1$, be a bounded connected open set. We consider the weighted eigenvalue problem $-\Delta u =\lambda m u$ in $\Omega$ with $\lambda \in \mathbb{R}$, $m\in L^\infty(\Omega)$ and with homogeneous Dirichlet and Robin boundary conditions. First, we study weak* continuity, convexity and G\^ateaux differentiability of the map $m\mapsto1/\lambda_1(m)$, where $\lambda_1(m)$ is the principal eigenvalue. Then, denoting by $\mathcal{G}(m_0)$ 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 $\lambda_1(m)$ over $\mathcal{G}(m_0)$. The minimization problem has been already discussed in some papers; here we prove some known results about the existence and characterization of minimizers of $\lambda_1(m)$. We underline that our approach allows us to treat 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 $\lambda_1(m)$ is more intricate than its minimization. In our work we discuss existence, uniqueness and characterization of maximizers both in $ \mathcal{G}(m_0)$ and in its weak* closure $\overline{\mathcal{G}(m_0)}$. In particular, we provide an original full description of the unique maximizer in the case of Dirichlet boundary conditions. In the context of the population dynamics, this kind of problems arise from the question of determining the optimal spatial location of favourable and unfavourable habitats in order to increase the chances of survival or extinction of a population.