{
  "id": "2108.05254",
  "version": "v1",
  "published": "2021-08-11T14:52:02.000Z",
  "updated": "2021-08-11T14:52:02.000Z",
  "title": "Optimal Shapes for Tree Roots",
  "authors": [
    "Alberto Bressan",
    "Sondre T. Galtung",
    "Qing Sun"
  ],
  "comment": "29 pages, 4 figures",
  "categories": [
    "math.OC"
  ],
  "abstract": "The paper studies a class of variational problems, modeling optimal shapes for tree roots. Given a measure $\\mu$ describing the distribution of root hair cells, we seek to maximize a harvest functional $\\mathcal{H}$, computing the total amount of water and nutrients gathered by the roots, subject to a cost for transporting these nutrients from the roots to the trunk. Earlier papers had established the existence of an optimal measure, and a priori bounds. Here we derive necessary conditions for optimality. Moreover, in space dimension $d=2$, we prove that the support of an optimal measure is nowhere dense.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-08-11T14:52:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35R06",
      "49Q10",
      "90B06",
      "92C80"
    ],
    "keywords": [
      "tree roots",
      "optimal measure",
      "root hair cells",
      "modeling optimal shapes",
      "variational problems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}