{
  "abstract": "We study the minimisation of the first solenoidal Maxwell eigenvalue $\\lambda_1^{\\mathrm{Max}}(\\Omega)$ in perfectly conducting cavities through the scale-invariant functional $J_{\\mathrm{Per}}(\\Omega):=\\mathrm{Per}(\\Omega)\\lambda_1^{\\mathrm{Max}}(\\Omega)$ in $\\mathbb R^3$. While the corresponding unconstrained problem is known to be ill-posed, we focus on bounded convex cavities and prove that $J_{\\mathrm{Per}}$ has a positive infimum in this class. The proof relies on the study of the three main possible behaviours for sequences of convex domains with bounded perimeter: convergence to a bounded convex domain with non-empty interior, collapse to a planar domain or collapse to a one-dimensional domain. The case of a planar collapse is handled through projection arguments, yielding a lower bound, while that of a one-dimensional collapse is controlled by a different argument, showing that in that case $J_{\\mathrm{Per}}$ must diverge to $+\\infty$. Although the question of existence of an optimal shape (that is, ruling out planar collapse) remains open, we derive several negative results and obstructions to existence; typically, no minimiser with a $\\mathscr C^{3,+}$ boundary exists. This relies on a fine analysis of first-order optimality conditions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.21794",
  "author_total": 4,
  "authors": [
    {
      "name": "Antoine Henrot"
    },
    {
      "name": "Idriss Mazari"
    },
    {
      "name": "Yannick Privat"
    },
    {
      "name": "Dominik Stantejsky"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fybzmj5d10q67.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.21794",
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  "id": "fybzmj5d10q67",
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  "landing_url": "https://hal.science/hal-05755062v1",
  "language": "en",
  "licence": "CC-BY-ND-4.0",
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  "metadata_licence": "CC0-1.0",
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      "math.AP"
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      "2610|Spectral Theory in Mathematical Physics"
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  "openalex": "W7213821374",
  "pdf_url": "https://hal.science/hal-05755062/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fybzmj5d10q67.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:36Z",
      "native_id": "hal-05755062",
      "source": "hal",
      "url": "https://hal.science/hal-05755062v1"
    },
    {
      "first_seen": "2026-10-30T12:23:07Z",
      "native_id": "2609.21794",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.21794"
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  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Shape optimisation of the first solenoidal Maxwell eigenvalue",
  "title_text": "Shape optimisation of the first solenoidal Maxwell eigenvalue",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:07Z",
  "url": "https://research.draconitic.com/paper/fybzmj5d10q67",
  "venue": "HAL",
  "versions": []
}