{
  "id": "2105.08434",
  "version": "v2",
  "published": "2021-05-18T10:54:07.000Z",
  "updated": "2021-05-31T18:24:41.000Z",
  "title": "Convergence of the Allen-Cahn equation with a nonlinear Robin Boundary Condition to Mean Curvature Flow with Contact Angle close to $90$°",
  "authors": [
    "Helmut Abels",
    "Maximilian Moser"
  ],
  "comment": "59 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "This paper is concerned with the sharp interface limit for the Allen-Cahn equation with a nonlinear Robin boundary condition in a bounded smooth domain $\\Omega\\subset\\mathbb{R}^2$. We assume that a diffuse interface already has developed and that it is in contact with the boundary $\\partial\\Omega$. The boundary condition is designed in such a way that the limit problem is given by the mean curvature flow with constant $\\alpha$-contact angle. For $\\alpha$ close to $90${\\deg} we prove a local in time convergence result for well-prepared initial data for times when a smooth solution to the limit problem exists. Based on the latter we construct a suitable curvilinear coordinate system and carry out a rigorous asymptotic expansion for the Allen-Cahn equation with the nonlinear Robin boundary condition. Moreover, we show a spectral estimate for the corresponding linearized Allen-Cahn operator and with its aid we derive strong norm estimates for the difference of the exact and approximate solutions using a Gronwall-type argument.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2021-05-31T18:24:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K57",
      "35B25",
      "35B36",
      "35R37"
    ],
    "keywords": [
      "nonlinear robin boundary condition",
      "mean curvature flow",
      "contact angle close",
      "allen-cahn equation",
      "convergence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 59,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}