{
  "id": "2106.00579",
  "version": "v1",
  "published": "2021-06-01T15:47:12.000Z",
  "updated": "2021-06-01T15:47:12.000Z",
  "title": "Yamabe systems, optimal partitions, and nodal solutions to the Yamabe equation",
  "authors": [
    "Mónica Clapp",
    "Angela Pistoia",
    "Hugo Tavares"
  ],
  "comment": "49 pages",
  "categories": [
    "math.AP",
    "math.DG"
  ],
  "abstract": "We give conditions for the existence of regular optimal partitions, with an arbitrary number $\\ell\\geq 2$ of components, for the Yamabe equation on a closed Riemannian manifold $(M,g)$. To this aim, we study a weakly coupled competitive elliptic system of $\\ell$ equations, related to the Yamabe equation. We show that this system has a least energy solution with nontrivial components if $\\dim M\\geq 10$, $(M,g)$ is not locally conformally flat and satisfies an additional geometric assumption whenever $\\dim M=10$. Moreover, we show that the limit profiles of the components of the solution separate spatially as the competition parameter goes to $-\\infty$, giving rise to an optimal partition. We show that this partition exhausts the whole manifold, and we prove the regularity of both the interfaces and the limit profiles, together with a free boundary condition. For $\\ell=2$ the optimal partition obtained yields a least energy sign-changing solution to the Yamabe equation with precisely two nodal domains.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-01T15:47:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B38",
      "35J20",
      "35J47",
      "35J60",
      "35R35",
      "49K20",
      "49Q10",
      "58J05"
    ],
    "keywords": [
      "yamabe equation",
      "nodal solutions",
      "yamabe systems",
      "limit profiles",
      "additional geometric assumption"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 49,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}