{
  "id": "2311.00218",
  "version": "v1",
  "published": "2023-11-01T01:37:10.000Z",
  "updated": "2023-11-01T01:37:10.000Z",
  "title": "On the solutions of nonlocal 1-Laplacian equation with $L^1$-data",
  "authors": [
    "Dingding Li",
    "Chao Zhang"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the solutions to a nonlocal 1-Laplacian equation given by $$ 2\\text{P.V.}\\int_{\\mathbb{R}^N}\\frac{u(x)-u(y)}{|u(x)-u(y)|} \\frac{dy}{|x-y|^{N+s}}=f(x) \\quad \\textmd{for } x\\in \\Omega, $$ with Dirichlet boundary condition $u(x)=0$ in $\\mathbb R^N\\backslash \\Omega$ and nonnegative $L^1$-data. By investigating the asymptotic behaviour of renormalized solutions $u_p$ to the nonlocal $p$-Laplacian equations as $p$ goes to $1^+$, we introduce a suitable definition of solutions and prove that the limit function $u$ of $\\{u_p\\}$ is a solution of the nonlocal $1$-Laplacian equation above.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-11-01T01:37:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "laplacian equation",
      "dirichlet boundary condition",
      "limit function",
      "asymptotic behaviour",
      "suitable definition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}