{
  "id": "2412.06437",
  "version": "v1",
  "published": "2024-12-09T12:26:45.000Z",
  "updated": "2024-12-09T12:26:45.000Z",
  "title": "Minimization of the first eigenvalue for the Lamé system",
  "authors": [
    "Antoine Henrot",
    "Antoine Lemenant",
    "Yannick Privat"
  ],
  "categories": [
    "math.AP",
    "math.SP"
  ],
  "abstract": "In this article, we address the problem of determining a domain in $\\mathbb{R}^N$ that minimizes the first eigenvalue of the Lam\\'e system under a volume constraint. We begin by establishing the existence of such an optimal domain within the class of quasi-open sets, showing that in the physically relevant dimensions $N = 2$ and $3$, the optimal domain is indeed an open set. Additionally, we derive both first and second-order optimality conditions. Leveraging these conditions, we demonstrate that in two dimensions, the disk cannot be the optimal shape when the Poisson ratio is below a specific threshold, whereas above this value, it serves as a local minimizer. We also extend our analysis to show that the disk is nonoptimal for Poisson ratios $\\nu$ satisfying $\\nu \\leq 0.4$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-12-09T12:26:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "49Q10",
      "47A10",
      "74B05",
      "74P10"
    ],
    "keywords": [
      "first eigenvalue",
      "poisson ratio",
      "minimization",
      "optimal domain",
      "second-order optimality conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}