  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 with
homogeneous Neumann boundary conditions. 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 $m_0$, under
the assumptions that $m_0$ is positive on a set of positive Lebesgue measure
and $\int_\Omega m\,dx<0$, we prove the existence and a characterization of
minimizers of $\lambda_1(m)$ and the non-existence of maximizers. Finally, we
show that, if $\Omega$ is a cylinder, then every minimizer is monotone with
respect to the direction of the generatrix. 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 for a
population to survive.
