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<p>Neumann eigenvalues having no mononicity property with respect to domain inclusion, it makes sense to study the two shape optimization problems min{<i>μ</i><sub><i>k</i></sub>(Ω): Ω convex, Ω ⊂ <i>D</i>,} (for a given box <i>D</i>) and max{<i>μ</i><sub><i>k</i></sub>(Ω): Ω convex, <i>ω</i> ⊂ Ω,} (for a given obstacle <i>ω</i>). In this paper, we study existence of a solution for these two problems in two dimensions and we give some qualitative properties. We also introduce the notion of self-domains that are domains solutions of these extremal problems for themselves and give examples of the disk and the square. A few numerical simulations are also presented.</p>

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Two extremum problems for Neumann eigenvalues

  • 期刊论文

  • 发表日期：

    2025年3月24日

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  • Lorenzo Cavallina,
  • Kei Funano,
  • Antoine Henrot,
  • Antoine Lemenant,
  • Ilaria Lucardesi,
  • Shigeru Sakaguchi

摘要

Neumann eigenvalues having no mononicity property with respect to domain inclusion, it makes sense to study the two shape optimization problems min{μk(Ω): Ω convex, Ω ⊂ D,} (for a given box D) and max{μk(Ω): Ω convex, ω ⊂ Ω,} (for a given obstacle ω). In this paper, we study existence of a solution for these two problems in two dimensions and we give some qualitative properties. We also introduce the notion of self-domains that are domains solutions of these extremal problems for themselves and give examples of the disk and the square. A few numerical simulations are also presented.