{
  "id": "2505.02260",
  "version": "v1",
  "published": "2025-05-04T21:31:53.000Z",
  "updated": "2025-05-04T21:31:53.000Z",
  "title": "Weighted minimum $α$-Green energy problems",
  "authors": [
    "Natalia Zorii"
  ],
  "comment": "21 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "For the $\\alpha$-Green kernel $g^\\alpha_D$ on a domain $D\\subset\\mathbb R^n$, $n\\geqslant2$, associated with the $\\alpha$-Riesz kernel $|x-y|^{\\alpha-n}$, where $\\alpha\\in(0,n)$ and $\\alpha\\leqslant2$, and a relatively closed set $F\\subset D$, we investigate the problem on minimizing the Gauss functional \\[\\int g^\\alpha_D(x,y)\\,d(\\mu\\otimes\\mu)(x,y)-2\\int g^\\alpha_D(x,y)\\,d(\\vartheta\\otimes\\mu)(x,y),\\] $\\vartheta$ being a given positive (Radon) measure concentrated on $D\\setminus F$, and $\\mu$ ranging over all probability measures of finite energy, supported in $D$ by $F$. For suitable $\\vartheta$, we find necessary and/or sufficient conditions for the existence of the solution to the problem, give a description of its support, provide various alternative characterizations, and prove convergence theorems when $F$ is approximated by partially ordered families of sets. The analysis performed is substantially based on the perfectness of the $\\alpha$-Green kernel, discovered by Fuglede and Zorii (Ann. Acad. Sci. Fenn. Math., 2018).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-05-04T21:31:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "31C15"
    ],
    "keywords": [
      "green energy problems",
      "weighted minimum",
      "green kernel",
      "riesz kernel",
      "gauss functional"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}