{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "DVAWCZWT3B"
    },
    {
      "field": "about",
      "target": "R8Y8DDTWBM"
    },
    {
      "field": "about",
      "target": "T6WEFSKYHZ"
    },
    {
      "field": "uses",
      "target": "R8Y8DDTWBM"
    },
    {
      "field": "uses",
      "target": "T6WEFSKYHZ"
    }
  ],
  "body": "\n## Summary\n\nThe Dirichlet-to-Neumann operator of a domain sends boundary values to normal derivatives.\nGiven a function $f$ on the boundary, it solves the [[Dirichlet problem|8D56GQ4AG8]] for [[Laplace's equation|S8RG6N1PRT]] with data $f$.\nThen it returns the normal derivative of the solution on the boundary; conductivity equations work the same way.\nIt is a basic example of a [[pseudo-differential operator|FF27N0BWTP]] on the boundary.\nIt carries the boundary measurements of electrical impedance tomography, in the [[Calderón problem|R8Y8DDTWBM]], and its eigenvalues are the Steklov eigenvalues.\n\n## Explanations\n\n### L1 human\n\nFix the temperature, or the voltage, on the boundary of a body and let it settle to equilibrium inside.\nThen measure how much heat, or current, flows out through each part of the boundary.\nThe Dirichlet-to-Neumann operator is the rule that turns the first list of numbers into the second.\nOn the unit disk, the boundary data $\\cos(k\\theta)$ extend to the harmonic function $r^k \\cos(k\\theta)$, whose radial derivative at $r = 1$ is $k \\cos(k\\theta)$.\nSo on the disk the operator multiplies the $k$-th Fourier mode by $\\abs{k}$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "35J25",
      "35R30",
      "35S05"
    ],
    "name": "Dirichlet-to-Neumann operator",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Poincar%C3%A9%E2%80%93Steklov_operator"
      }
    ],
    "tag": "W783KDGT8E"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/W783KDGT8E/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/W783KDGT8E.md",
  "msc": [
    "35J25",
    "35R30",
    "35S05"
  ],
  "name": "Dirichlet-to-Neumann operator",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/W78/W783KDGT8E.md",
  "status": null,
  "tag": "W783KDGT8E",
  "uri": "m2://W783KDGT8E",
  "url": "https://tamnd.github.io/math2/wiki/dirichlet-to-neumann-operator/",
  "writing": {
    "tag": "W783KDGT8E",
    "kind": "concept",
    "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
      }
    ]
  }
}