{
  "id": "1806.02065",
  "version": "v1",
  "published": "2018-06-06T08:44:44.000Z",
  "updated": "2018-06-06T08:44:44.000Z",
  "title": "Convergence of the Allen-Cahn Equation to the Mean Curvature Flow with $90^\\circ$-Contact Angle in 2D",
  "authors": [
    "Helmut Abels",
    "Maximilian Moser"
  ],
  "comment": "53 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the sharp interface limit of the Allen-Cahn equation with homogeneous Neumann boundary condition in a two-dimensional domain $\\Omega$, in the situation where an interface has developed and intersects $\\partial\\Omega$. Here a parameter $\\varepsilon>0$ in the equation, which is related to the thickness of the diffuse interface, is sent to zero. The limit problem is given by mean curvature flow with a $90$\\textdegree-contact angle condition and convergence using strong norms is shown for small times. Here we assume that a smooth solution to this limit problem exists on $[0,T]$ for some $T>0$ and that it can be parametrized suitably. With the aid of asymptotic expansions we construct an approximate solution for the Allen-Cahn equation and estimate the difference of the exact and approximate solution with the aid of a spectral estimate for the linearized Allen-Cahn operator.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-06-06T08:44:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K57",
      "35B25",
      "35B36",
      "35R37"
    ],
    "keywords": [
      "mean curvature flow",
      "allen-cahn equation",
      "convergence",
      "limit problem",
      "approximate solution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 53,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}