Abstract (EN)

Let m be a bounded function and α a nonnegative parameter. This article is concerned with the first eigenvalue λα(m) of the drifted Laplacian type operator Lm given by Lm(u)=−div((1+αm)∇u)−mu on a smooth bounded domain, with Dirichlet boundary conditions. Assuming uniform pointwise and integral bounds on m, we investigate the issue of minimizing λα(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.