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

TL;DR

This paper studies the optimal spatial design of a favorable habitat in a periodically fragmented environment, formulated as minimizing the principal eigenvalue $\lambda_1(m)$ of the periodic Laplacian with a bang-bang indefinite weight $m=\mathds{1}_D - \beta \mathds{1}_{\mathcal{C}\setminus D}$ under a fixed volume constraint. The authors develop a singular perturbation/blow-up analysis to compare the periodic problem with the limit problem on $\mathbb{R}^N$, establishing that the optimal region concentrates to a ball and that the eigenfunction converges to the radial profile $w$ solving the limit eigenproblem. They obtain a first-order expansion $\Lambda(\delta)=\delta^{-2/N} \Lambda_0 + o(\delta^{-2/N})$ with exponential corrections, and prove that the free boundary is nearly spherical with $C^{1,1}$ regularity and $C^{1,\alpha}$ decay rates for any $0<\alpha<1$. These results extend existing small-volume analyses to the periodic setting, providing sharp quantitative convergence rates for the optimal sets and offering insights into habitat optimization and persistence thresholds.

Abstract

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 $C^{1,α}$ sense, for every $α<1$.

Theorems & Definitions (36)

• Proposition 1.1
• Theorem 1.2
• Remark 1.3
• Definition 1.4
• Corollary 1.5
• Theorem 1.6
• Remark 1.7
• Remark 1.8
• Remark 1.9
• Remark 1.10