{
  "id": "1810.07943",
  "version": "v1",
  "published": "2018-10-18T08:29:50.000Z",
  "updated": "2018-10-18T08:29:50.000Z",
  "title": "Existence and regularity of optimal shapes for elliptic operators with drift",
  "authors": [
    "Emmanuel Russ",
    "Baptiste Trey",
    "Bozhidar Velichkov"
  ],
  "categories": [
    "math.AP",
    "math.OC"
  ],
  "abstract": "This paper is devoted to the study of shape optimization problems for the first eigenvalue of the elliptic operator with drift $L = -\\Delta + V(x) \\cdot \\nabla$ with Dirichlet boundary conditions, where $V$ is a bounded vector field. In the first instance, we prove the existence of a principal eigenvalue $\\lambda_1(\\Omega,V)$ for a bounded quasi-open set $\\Omega$ which enjoys similar properties to the case of open sets. Then, given $m>0$ and $\\tau\\geq 0$, we show that the minimum of the following non-variational problem \\begin{equation*} \\min\\Big\\{\\lambda_1(\\Omega,V)\\ :\\ \\Omega\\subset D\\ \\text{quasi-open},\\ |\\Omega|\\leq m,\\ \\|V\\|_{L^\\infty}\\le \\tau\\Big\\}. \\end{equation*} is achieved, where the box $D\\subset \\mathbb{R}^d$ is a bounded open set. The existence when $V$ is fixed, as well as when $V$ varies among all the vector fields which are the gradient of a Lipschitz function, are also proved. The second interest and main result of this paper is the regularity of the optimal shape $\\Omega^\\ast$ solving the minimization problem \\begin{equation*} \\min\\Big\\{\\lambda_1(\\Omega,\\nabla\\Phi)\\ :\\ \\Omega\\subset D\\ \\text{quasi-open},\\ |\\Omega|\\leq m\\Big\\}, \\end{equation*} where $\\Phi$ is a given Lipschitz function on $D$. We prove that the topological boundary $\\partial\\Omega^\\ast$ is composed of a {\\it regular part} which is locally the graph of a $C^{1,\\alpha}$ function and a {\\it singular part} which is empty if $d<d^\\ast$, discrete if $d=d^\\ast$ and of locally finite $\\mathcal{H}^{d-d^\\ast}$ Hausdorff measure if $d>d^\\ast$, where $d^\\ast \\in \\{5,6,7\\}$ is the smallest dimension at which there exists a global solution to the one-phase free boundary problem with singularities. Moreover, if $D$ is smooth, we prove that, for each $x\\in \\partial\\Omega^{\\ast}\\cap \\partial D$, $\\partial\\Omega^\\ast$ is $C^{1,\\alpha}$ in a neighborhood of $x$, for some $\\alpha\\leq \\frac 12$. This last result is optimal in the sense that $C^{1,1/2}$ is the best regularity that one can expect.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-18T08:29:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "elliptic operator",
      "optimal shape",
      "regularity",
      "open set",
      "lipschitz function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}