{
  "id": "1605.00870",
  "version": "v1",
  "published": "2016-04-30T18:04:18.000Z",
  "updated": "2016-04-30T18:04:18.000Z",
  "title": "A study on class of elliptic PDEs with measure data",
  "authors": [
    "Ratan Kr Giri",
    "Debajyoti Choudhuri"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the following PDE \\begin{align*} \\begin{split} -\\Delta_p u + g\\circ u & = \\mu\\,\\,\\mbox{in}\\,\\,\\Omega,\\\\ u & = \\nu\\,\\,\\mbox{on}\\,\\,\\partial\\Omega, \\end{split} \\end{align*} where $\\Omega$ is a bounded domain $\\mathbb{R}^N$ with $\\partial\\Omega$ being the boundary of $\\Omega$, $1<p<N$. The problem investigated here is motivated from the work of Bhakta and Marcus \\cite{bhakta} where a problem close to the problem above has been addressed. Suppose $(\\mu_n)$, $(\\nu_n)$ are a sequence of measures for which the above PDE has a corresponding sequence of solutions say $(u_n)$. In general if $\\mu_n$, $\\nu_n$ converges weakly to $\\mu$, $\\nu$ respectively and the corresponding sequence of solutions $u_n$ converges to $u$ in $L^1(\\Omega)$, then $u$ need not be a solution to the above equation. However it is found in the literature for $p=2$ that to each pair of measures $(\\mu,\\nu)$ there exists another pair of measures, say, $(\\mu^{\\#},\\nu^{\\#})$, called as the {\\it reduced limit} corresponding to which $u$ is a solution to the above equation. We address a similar issue for the case of $p \\neq 2$, $p < N$. Further, we also derive a similar relationship between the reduced and the weak limit of the measure as in \\cite{bhakta}, but for $p \\neq 2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-04-30T18:04:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "elliptic pdes",
      "measure data",
      "corresponding sequence",
      "problem close",
      "similar relationship"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160500870G"
    }
  }
}