{
  "id": "2403.07793",
  "version": "v1",
  "published": "2024-03-12T16:32:25.000Z",
  "updated": "2024-03-12T16:32:25.000Z",
  "title": "Optimal regularity for nonlocal elliptic equations and free boundary problems",
  "authors": [
    "Xavier Ros-Oton",
    "Marvin Weidner"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this article we establish for the first time the $C^s$ boundary regularity of solutions to nonlocal elliptic equations with kernels $K(y)\\asymp |y|^{-n-2s}$. This was known to hold only when $K$ is homogeneous, and it is quite surprising that it holds for general inhomogeneous kernels, too. As an application of our results, we also establish the optimal $C^{1+s}$ regularity of solutions to obstacle problems for general nonlocal operators with kernels $K(y)\\asymp |y|^{-n-2s}$. Again, this was only known when $K$ is homogeneous, and it solves a long-standing open question in the field. A new key idea is to construct a 1D solution as a minimizer of an appropriate nonlocal one-phase free boundary problem, for which we establish optimal $C^s$ regularity and non-degeneracy estimates.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-03-12T16:32:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47G20",
      "35B65",
      "31B05",
      "60J75",
      "35K90"
    ],
    "keywords": [
      "nonlocal elliptic equations",
      "optimal regularity",
      "appropriate nonlocal one-phase free boundary",
      "nonlocal one-phase free boundary problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}