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<p>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 into 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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(C^{1,1}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(C^{1,\alpha }\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>α</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> sense, for every <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\alpha &lt;1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>&lt;</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

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Shape Optimization of a Small Favorable Region in a Periodically Fragmented Environment

  • 期刊论文

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  • 发表日期：

    2026年3月17日

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  • Gianmaria Verzini

摘要

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 into 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 \(C^{1,1}\) 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 \(C^{1,\alpha }\) C 1 , α sense, for every \(\alpha <1\) α < 1 .