{
  "id": "1405.3329",
  "version": "v2",
  "published": "2014-05-13T23:23:23.000Z",
  "updated": "2016-10-06T08:09:32.000Z",
  "title": "The Dirichlet problem for elliptic systems with data in Köthe function spaces",
  "authors": [
    "José María Martell",
    "Dorina Mitrea",
    "Irina Mitrea",
    "Marius Mitrea"
  ],
  "categories": [
    "math.AP",
    "math.CA",
    "math.FA"
  ],
  "abstract": "We show that the boundedness of the Hardy-Littlewood maximal operator on a K\\\"othe function space ${\\mathbb{X}}$ and on its K\\\"othe dual ${\\mathbb{X}}'$ is equivalent to the well-posedness of the $\\mathbb{X}$-Dirichlet and $\\mathbb{X}'$-Dirichlet problems in $\\mathbb{R}^{n}_{+}$ in the class of all second-order, homogeneous, elliptic systems, with constant complex coefficients. As a consequence, we obtain that the Dirichlet problem for such systems is well-posed for boundary data in Lebesgue spaces, variable exponent Lebesgue spaces, Lorentz spaces, Zygmund spaces, as well as their weighted versions. We also discuss a version of the aforementioned result which contains, as a particular case, the Dirichlet problem for elliptic systems with data in the classical Hardy space $H^1$, and the Beurling-Hardy space ${\\rm HA}^p$ for $p\\in(1,\\infty)$. Based on the well-posedness of the $L^p$-Dirichlet problem we then prove the uniqueness of the Poisson kernel associated with such systems, as well as the fact that they generate a strongly continuous semigroup in natural settings. Finally, we establish a general Fatou type theorem guaranteeing the existence of the pointwise nontangential boundary trace for null-solutions of such systems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-05-13T23:23:23.000Z",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2016-10-06T08:09:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35C15",
      "35J57",
      "42B37",
      "46E30",
      "35B65",
      "35E05",
      "42B25",
      "42B30",
      "42B35",
      "74B05"
    ],
    "keywords": [
      "dirichlet problem",
      "elliptic systems",
      "köthe function spaces",
      "nontangential boundary trace",
      "hardy-littlewood maximal operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.3329M"
    }
  }
}