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

## TL;DR

This paper investigates the relationship between two spectral shape optimization problems, focusing on a polygonal domain, and provides quantitative estimates for their solutions and eigenvalues as the outside environment becomes more hostile.

## Contribution

It refines previous analysis by offering quantitative estimates for the convergence of optimal levels and eigenvalues in a polygonal setting, advancing understanding of singular perturbations in shape optimization.

## Key findings

- Quantitative estimates of optimal level convergence in polygonal domains.
- Eigenvalue convergence analysis under singular perturbations.
- Enhanced understanding of spectral shape optimization in polygons.

## Abstract

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 $\Omega$; 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 $\Omega$. 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 $\Omega$ is a planar polygon, providing quantitative estimates of the optimal level convergence, as well as of the involved eigenvalues.