Shape optimization of a small favorable region in a periodically fragmented environment

We consider a shape optimization problem for the persistence threshold of a biological species dispersing in a periodically fragmented environment, the unknown shape corresponding to the portion of the habitat which is favorable to the population. Analytically, this translates in the minimization of a weighted eigenvalue of the periodic Laplacian, with respect to a bang-bang indefinite weight. For such problem, we exploit some recent results obtained in the framework of Dirichlet or Neumann boundary conditions, to provide a full description of the singularly perturbed regime in which the volume of the favorable zone vanishes. First, we show that the optimal favorable zone shrinks to a connected, convex, nearly spherical set, in $C^{1,1}$ sense. Secondly, we show that the spherical asymmetry of the optimal favorable zone decays exponentially, with respect to a negative power of its volume, in the $C^{1,\alpha }$ sense, for every $\alpha <1$.