Calculus of Variations and Geometric Measure Theory
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D. Mazzoleni - B. Pellacci - G. Verzini

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

created by mazzoleni on 19 Feb 2019

[BibTeX]

Preprint

Inserted: 19 feb 2019
Last Updated: 19 feb 2019

Year: 2019

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.


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