{
  "id": "2411.09930",
  "version": "v1",
  "published": "2024-11-15T04:13:50.000Z",
  "updated": "2024-11-15T04:13:50.000Z",
  "title": "On some regularity properties of mixed local and nonlocal elliptic equations",
  "authors": [
    "Xifeng Su",
    "Enrico Valdinoci",
    "Yuanhong Wei",
    "Jiwen Zhang"
  ],
  "comment": "Journal of Differential Equations",
  "categories": [
    "math.AP"
  ],
  "abstract": "This article is concerned with ``up to $C^{2, \\alpha}$-regularity results'' about a mixed local-nonlocal nonlinear elliptic equation which is driven by the superposition of Laplacian and fractional Laplacian operators. First of all, an estimate on the $L^\\infty$ norm of weak solutions is established for more general cases than the ones present in the literature, including here critical nonlinearities. We then prove the interior $C^{1,\\alpha}$-regularity and the $C^{1,\\alpha}$-regularity up to the boundary of weak solutions, which extends previous results by the authors [X. Su, E. Valdinoci, Y. Wei and J. Zhang, Math. Z. (2022)], where the nonlinearities considered were of subcritical type. In addition, we establish the interior $C^{2,\\alpha}$-regularity of solutions for all $s\\in(0,1)$ and the $C^{2,\\alpha}$-regularity up to the boundary for all $s\\in(0,\\frac{1}{2})$, with sharp regularity exponents. For further perusal, we also include a strong maximum principle and some properties about the principal eigenvalue.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-11-15T04:13:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B65",
      "35R11",
      "35J67"
    ],
    "keywords": [
      "nonlocal elliptic equations",
      "regularity properties",
      "mixed local-nonlocal nonlinear elliptic equation",
      "weak solutions",
      "fractional laplacian operators"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}