{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $U \\subset \\R^n$ be a bounded open set with $C^1$ boundary $\\partial U$, carrying its surface measure, and let $1 \\le p \u003c \\infty$.\nThe *trace operator* is the unique bounded linear map\n$$T \\colon W^{1,p}(U) \\to L^p(\\partial U)$$\nsuch that $T u = u|_{\\partial U}$ for every $u \\in W^{1,p}(U)$ that is continuous on $\\overline{U}$.\nHere $W^{1,p}(U)$ is the [[Sobolev space|M4CEZ5K9HP]].\nThe function $T u$ is the *trace* of $u$ on $\\partial U$.\n\n## Remarks\n\nThe operator exists and is unique.\nFor $u$ smooth up to the boundary, the divergence theorem gives $\\norm{u|_{\\partial U}}_{L^p(\\partial U)} \\le C \\norm{u}_{W^{1,p}(U)}$ with $C$ depending on $p$ and $U$.\nSuch functions are dense in $W^{1,p}(U)$, so restriction extends uniquely to a [[continuous linear map|03TM135G0N]] on all of $W^{1,p}(U)$.\n\nExample: on $U = (0, 1)$ the boundary is $\\set{0, 1}$, and every $u \\in W^{1,p}(0, 1)$ has a continuous representative.\nIts trace is the pair of values $(u(0), u(1))$ of that representative.\n\nNon-example: on $U = (0, 1)$, restriction to the boundary is not bounded for the $L^p$ norm alone.\nThe functions $u_k(x) = (1 - x)^k$ satisfy $u_k(0) = 1$ while $\\norm{u_k}_{L^p(0,1)} \\to 0$, so no bounded map $L^p(0, 1) \\to \\R^2$ extends restriction.\nThis is why the definition uses $W^{1,p}(U)$.\n\nThe kernel of $T$ is $W^{1,p}_0(U)$.\nFor $1 \u003c p \u003c \\infty$, the image of $T$ is the [[fractional Sobolev space|N6NX1VT837]] $W^{1 - 1/p, p}(\\partial U)$, a result of Gagliardo.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "Z2T5R88QDB",
    "kind": "definition",
    "msc": [
      "46E35"
    ],
    "name": "Trace operator",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Trace_operator"
      }
    ],
    "tag": "KEY8MM7F28",
    "uses": [
      "M4CEZ5K9HP",
      "SASP14MA1Z",
      "03TM135G0N",
      "N6NX1VT837"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/KEY8MM7F28/",
  "links": [
    {
      "field": "defines",
      "target": "Z2T5R88QDB"
    },
    {
      "field": "uses",
      "target": "M4CEZ5K9HP"
    },
    {
      "field": "uses",
      "target": "SASP14MA1Z"
    },
    {
      "field": "uses",
      "target": "03TM135G0N"
    },
    {
      "field": "uses",
      "target": "N6NX1VT837"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/KEY8MM7F28.md",
  "msc": [
    "46E35"
  ],
  "name": "Trace operator",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/KEY/KEY8MM7F28.md",
  "status": null,
  "tag": "KEY8MM7F28",
  "uri": "m2://KEY8MM7F28",
  "url": "https://tamnd.github.io/math2/wiki/trace-operator-definition/",
  "writing": {
    "tag": "KEY8MM7F28",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}