{
  "id": "2004.09782",
  "version": "v1",
  "published": "2020-04-21T07:17:42.000Z",
  "updated": "2020-04-21T07:17:42.000Z",
  "title": "The diamagnetic inequality for the Dirichlet-to-Neumann operator",
  "authors": [
    ". A. F. M. ter Elst",
    "El Maati Ouhabaz"
  ],
  "categories": [
    "math.AP",
    "math.FA"
  ],
  "abstract": "Let $\\Omega$ be a bounded domain in R d with Lipschitz boundary $\\Gamma$. We define the Dirichlet-to-Neumann operator N on L 2 ($\\Gamma$) associated with a second order elliptic operator A = -- d k,j=1 $\\partial$ k (c kl $\\partial$ l) + d k=1 b k $\\partial$ k -- $\\partial$ k (c k $\\times$) + a 0. We prove a criterion for invariance of a closed convex set under the action of the semigroup of N. Roughly speaking, it says that if the semigroup generated by --A, endowed with Neumann boundary conditions, leaves invariant a closed convex set of L 2 ($\\Omega$), then the 'trace' of this convex set is invariant for the semigroup of N. We use this invariance to prove a criterion for the domination of semigroups of two Dirichlet-to-Neumann operators. We apply this criterion to prove the diamagnetic inequality for such operators on L 2 ($\\Gamma$).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-04-21T07:17:42.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "dirichlet-to-neumann operator",
      "diamagnetic inequality",
      "closed convex set",
      "second order elliptic operator",
      "neumann boundary conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}