{
  "id": "1402.3320",
  "version": "v2",
  "published": "2014-02-13T21:58:33.000Z",
  "updated": "2016-09-18T16:38:27.000Z",
  "title": "Rate of Convergence for Large Coupling Limits in Sobolev Spaces",
  "authors": [
    "Ikemefuna Agbanusi"
  ],
  "comment": "The paper contains 1 Figure, has been shortened to 10 pages and also has updated References. This version is close to the Journal (Comm. P.D.E.) accepted version and incorprates some comments from a reviewer. Any other comments on the content of the paper are very welcome",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We estimate the rate of convergence, in the so-called large coupling limit, for Schr\\\"odinger type operators on bounded domains. The Schr\\\"odinger we deal with have \"interaction potentials\" supported in a compact inclusion. We show that if the boundary of the inclusion is sufficiently smooth, one essentially recovers the \"free Hamiltonian\" in the exterior domain with Dirichlet boundary conditions. In addition, we obtain a convergence rate, in $L^2$, that is $\\mathcal{O}(\\lambda^{-\\frac{1}{4}})$ where $\\lambda$ is the coupling parameter. Our methods include energy estimates, trace estimates, interpolation and duality.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-02-13T21:58:33.000Z",
      "abstract": "We obtain estimates for convergence rates, in a scale of anisotropic Sobolev spaces, for certain singularly perturbed parabolic problems. The equations we consider correspond to the two primary spatially continuous models used in modelling stochastic reaction diffusion. One model, due to Smoluchowskii, uses a Dirichlet boundary condition to capture the reaction mechanism while the other model uses an interaction potential with a coupling constant $\\lambda$ and corresponds to a Schr\\\"odinger type operator. Thus we show that the two models agree in the large coupling limit and estimate the rate of convergence.",
      "comment": "19 pages, 1 Figure, 14 References. Any comments on the content of the paper are very welcome",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2016-09-18T16:38:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35R05",
      "35B25",
      "58J35"
    ],
    "keywords": [
      "large coupling limit",
      "convergence",
      "anisotropic sobolev spaces",
      "dirichlet boundary condition",
      "modelling stochastic reaction diffusion"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1402.3320A"
    }
  }
}