  We study the optimization of the positive principal eigenvalue of an
indefinite weighted problem, associated with the Neumann Laplacian in a box
$\Omega\subset\mathbb{R}^N$, which arises in the investigation of the survival
threshold in population dynamics. When trying to minimize such eigenvalue with
respect to the weight, one is lead to consider a shape optimization problem,
which is known to admit no spherical optimal shapes (despite some previously
stated conjectures). We investigate whether spherical shapes can be recovered
in some singular perturbation limit. More precisely we show that, whenever the
negative part of the weight diverges, the above shape optimization problem
approaches in the limit the so called spectral drop problem, which involves the
minimization of the first eigenvalue of the mixed Dirichlet-Neumann Laplacian.
We prove that, for suitable choices of the box $\Omega$, the optimal shapes for
this second problem are indeed spherical; moreover, for general $\Omega$, we
show that small volume spectral drops are asymptotically spherical, with center
at points of $\partial\Omega$ having large mean curvature.

Analysis of PDEs