{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "1X62YMAGMN"
    },
    {
      "field": "uses",
      "target": "4PK2T949SQ"
    },
    {
      "field": "uses",
      "target": "B51J3BC610"
    },
    {
      "field": "uses",
      "target": "CCEKM4241P"
    },
    {
      "field": "uses",
      "target": "EFFC7688KR"
    },
    {
      "field": "uses",
      "target": "STMT1K1865"
    },
    {
      "field": "uses",
      "target": "W6SXM59TS4"
    },
    {
      "field": "defines",
      "target": "3F4Y6NEVM4"
    },
    {
      "field": "about",
      "target": "B51J3BC610"
    },
    {
      "field": "about",
      "target": "STMT1K1865"
    },
    {
      "field": "about",
      "target": "W6SXM59TS4"
    }
  ],
  "body": "\n## Summary\n\nA boundary point of a bounded open set is regular if the generalized solution of the [[Dirichlet problem|8D56GQ4AG8]] tends to the prescribed value there.\nThis must hold for every continuous boundary function.\nThe Dirichlet problem is solvable for all continuous data exactly when every boundary point is regular.\nRegularity is a local property: it holds at $\\xi$ if and only if there is a [[barrier|ED0SZS1VED]] at $\\xi$.\nIt is characterized by Wiener's criterion in terms of capacity, and the irregular points always form a polar set.\n\n## Explanations\n\n### L1 human\n\nAt most boundary points the solution of the Dirichlet problem behaves well and takes the boundary value you prescribe.\nSome points are so small or so thin that the solution does not notice them.\nThe centre of a punctured disc is the simplest such irregular point: whatever value you prescribe there, the solution ignores it.\nA point on a smooth piece of boundary, or the corner of a square, is regular.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "31B25",
      "31B20",
      "31A25"
    ],
    "name": "Regular boundary point",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Perron_method"
      }
    ],
    "tag": "B7948M365K"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/B7948M365K/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/B7948M365K.md",
  "msc": [
    "31B25",
    "31B20",
    "31A25"
  ],
  "name": "Regular boundary point",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/B79/B7948M365K.md",
  "status": null,
  "tag": "B7948M365K",
  "uri": "m2://B7948M365K",
  "url": "https://tamnd.github.io/math2/wiki/regular-boundary-point/",
  "writing": {
    "tag": "B7948M365K",
    "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
      }
    ]
  }
}