Quantitative analysis of a singularly perturbed shape optimization problem in a polygon

We carry on our study of the connection between two shape optimization problems with spectral cost. On the one hand, we consider the optimal design problem for the survival threshold of a population living in a heterogenous habitat $\mathrm{\Omega }$; this problem arises when searching for the optimal shape and location of a shelter zone in order to prevent extinction of the species. On the other hand, we deal with the spectral drop problem, which consists in minimizing a mixed Dirichlet-Neumann eigenvalue in a box $\mathrm{\Omega }$. In a previous paper arXiv:1811.01623 we proved that the latter one can be obtained as a singular perturbation of the former, when the region outside the refuge is more and more hostile. In this paper we sharpen our analysis in case $\mathrm{\Omega }$ is a planar polygon, providing quantitative estimates of the optimal level convergence, as well as of the involved eigenvalues.