  Let $m$ be a bounded function and $\alpha$ a nonnegative parameter. This
article is concerned with the first eigenvalue $\lambda\_\alpha(m)$ of the
drifted Laplacian type operator $\mathcal L\_m$ given by $\mathcal L\_m(u)=
-\operatorname{div} \left((1+\alpha m)\nabla u\right)-mu$ on a smooth bounded
domain, with Dirichlet boundary conditions. Assuming uniform pointwise and
integral bounds on $m$, we investigate the issue of minimizing
$\lambda\_\alpha(m)$ with respect to $m$. Such a problem is related to the
so-called "two phase extremal eigenvalue problem" and arises naturally, for
instance in population dynamics where it is related to the survival ability of
a species in a domain. We prove that unless the domain is a ball, this problem
has no "regular" solution. We then provide a careful analysis in the case of a
ball by: (1) characterizing the solution among all radially symmetric resources
distributions, with the help of a new method involving a homogenized version of
the problem; (2) proving in a more general setting, a stability result for the
centered distribution of resources with the help of a monotonicity principle
for second order shape derivatives which significantly simplifies the analysis.
