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Let $\Omega\subset\mathbb{R}^N$, $N\geq 1$, be an open bounded connected set.We consider the indefinite weighted eigenvalue problem $-\Delta u =\lambda m u$in $\Omega$ with $\lambda \in \mathbb{R}$, $m\in L^\infty(\Omega)$ and withhomogeneous Neumann boundary conditions. We study weak* continuity, convexityand 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 $m_0$, underthe assumptions that $m_0$ is positive on a set of positive Lebesgue measureand $\int_\Omega m\,dx<0$, we prove the existence and a characterization ofminimizers of $\lambda_1(m)$ and the non-existence of maximizers. Finally, weshow that, if $\Omega$ is a cylinder, then every minimizer is monotone withrespect to the direction of the generatrix. In the context of the populationdynamics, this kind of problems arise from the question of determining theoptimal spatial location of favourable and unfavourable habitats for apopulation to survive.

Optimization of the principal eigenvalue of the Neumann Laplacian with indefinite weight and monotonicity of minimizers in cylinders