Shape Optimization of a Weighted Two-Phase Dirichlet Eigenvalue

This article is concerned with a spectral optimization problem: in a smooth bounded domain Omega, for a bounded function m and a nonnegative parameter alpha, consider the first eigenvalue lambda(alpha)(m) of the operator L-m given by L-m(u) = - div (1 + alpha m) del u) - mu. 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.