{
  "id": "2412.20335",
  "version": "v1",
  "published": "2024-12-29T03:30:18.000Z",
  "updated": "2024-12-29T03:30:18.000Z",
  "title": "Flat level sets of Allen-Cahn equation in half-space",
  "authors": [
    "Wenkui Du",
    "Ling Wang",
    "Yang Yang"
  ],
  "comment": "13 pages, 2 figures",
  "categories": [
    "math.AP"
  ],
  "abstract": "We prove a half-space Bernstein theorem for Allen-Cahn equation. More precisely, we show that every solution $u$ of the Allen-Cahn equation in the half-space $\\overline{\\mathbb{R}^n_+}:=\\{(x_1,x_2,\\cdots,x_n)\\in\\mathbb{R}^n:\\,x_1\\geq 0\\}$ with $|u|\\leq 1$, boundary value given by the restriction of a one-dimensional solution on $\\{x_1=0\\}$ and monotone condition $\\partial_{x_n}u>0$ as well as limiting condition $\\lim_{x_n\\to\\pm\\infty}u(x',x_n)=\\pm 1$ must itself be one-dimensional, and the parallel flat level sets and $\\{x_1=0\\}$ intersect at the same fixed angle in $(0, \\frac{\\pi}{2}]$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-12-29T03:30:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "allen-cahn equation",
      "parallel flat level sets",
      "half-space bernstein theorem",
      "boundary value",
      "monotone condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}