{
  "id": "2109.11267",
  "version": "v1",
  "published": "2021-09-23T10:12:14.000Z",
  "updated": "2021-09-23T10:12:14.000Z",
  "title": "Scattering by finely-layered obstacles: frequency-explicit bounds and homogenization",
  "authors": [
    "Théophile Chaumont-Frelet",
    "Euan A. Spence"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the scalar Helmholtz equation with variable, discontinuous coefficients, modelling transmission of acoustic waves through an anisotropic penetrable obstacle. We first prove a well-posedness result and a frequency-explicit bound on the solution operator, with both valid for sufficiently-large frequency and for a class of coefficients that satisfy certain monotonicity conditions in one spatial direction, and are only assumed to be bounded (i.e., $L^\\infty$) in the other spatial directions. This class of coefficients therefore includes coefficients modelling transmission by penetrable obstacles with a (potentially large) number of layers (in 2-d) or fibres (in 3-d). Importantly, the frequency-explicit bound holds uniformly for all coefficients in this class; this uniformity allows us to consider highly-oscillatory coefficients and study the limiting behaviour when the period of oscillations goes to zero. In particular, we bound the $H^1$ error committed by the first-order bulk correction to the homogenized transmission problem, with this bound explicit in both the period of oscillations of the coefficients and the frequency of the Helmholtz equation; to our knowledge, this is the first homogenization result for the Helmholtz equation that is explicit in these two quantities and valid without the assumption that the frequency is small.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-09-23T10:12:14.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "coefficients",
      "finely-layered obstacles",
      "spatial direction",
      "modelling transmission",
      "penetrable obstacle"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}