{
  "id": "1612.08402",
  "version": "v1",
  "published": "2016-12-26T15:23:28.000Z",
  "updated": "2016-12-26T15:23:28.000Z",
  "title": "1-Laplacian Type Equations with Neumann Boundary Condition",
  "authors": [
    "Amir Moradifam"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We study existence of solutions of the general 1-Laplacian type equation \\begin{eqnarray}\\label{abs} \\left\\{ \\begin{array}{ll} \\nabla_x \\cdot \\nabla_p \\varphi(x,\\frac{Du}{|Du|})=0 &\\text{in } \\Omega \\left[ \\nabla_p \\varphi (x, \\frac{Du}{|Du|}), \\nu_{\\Omega} \\right]=\\lambda g&\\text{on }\\partial \\Omega, \\end{array} \\right. \\end{eqnarray} where $\\varphi(x,p)$ is convex, continuous, and homogeneous function of degree $1$ with respect to the $p$ variable, and $g$ satisfies the comparability condition $\\int_{\\partial \\Omega} g dS=0$. We prove that for every $0\\not \\equiv g \\in L^{\\infty}(\\partial \\Omega)$ there exists a unique $\\lambda^*>0$ for which the equation (\\ref{abs}) has infinitely many solutions in $BV(\\Omega)$, and if $\\lambda \\neq \\lambda^*$, then (\\ref{abs}) does not admit any solutions. In addition, when $\\lambda=\\lambda^*$, it is shown that there exists a divergence free vector field $T\\in (L^{\\infty}(\\Omega))^n$ that determines the structure of level sets of all solutions of (\\ref{abs}), i.e. $T$ determines $\\frac{Du}{|Du|}$, $|Du|-$ a.e. in $\\Omega$, for all solutions $u$. We also present a numerical algorithm which simultaneously finds $\\lambda^*$, $T$, and a solution of the degenerate equation (\\ref{abs}).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-12-26T15:23:28.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "neumann boundary condition",
      "type equation",
      "divergence free vector field",
      "level sets",
      "determines"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}