{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "SSYPTQC4C1"
    }
  ],
  "body": "\n## Summary\n\nAn elliptic boundary value problem asks for a solution of an elliptic equation in a domain, subject to conditions on the boundary of the domain.\nThe model case is the Dirichlet problem for the [[Laplacian|0E0B6RA8K0]]: $\\Delta u = f$ inside and $u = g$ on the boundary.\nIn general, the operator must be elliptic and the boundary conditions must satisfy the [[Lopatinskii–Shapiro condition|48MEJEMEXM]].\nSuch problems have a priori estimates, regularity up to the boundary and a Fredholm solution theory.\n\n## Explanations\n\n### L1 human\n\nThink of the steady temperature in a metal plate whose edge is held at given temperatures.\nInside, heat flow balances out, which gives an elliptic equation; on the edge, the temperature is prescribed.\nTogether these determine the temperature everywhere.\nElliptic boundary value problems are the general form of this kind of question, for equilibrium states in physics and geometry.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "35J25",
      "35J40"
    ],
    "name": "Elliptic boundary value problem",
    "same_as": {
      "wikidata": "Q5365788",
      "wikipedia": "Elliptic boundary value problem"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Elliptic_boundary_value_problem"
      }
    ],
    "specializes": [
      "TM8VQJWGHZ"
    ],
    "tag": "C1F7T523G0"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/C1F7T523G0/",
  "links": [
    {
      "field": "specializes",
      "target": "TM8VQJWGHZ"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/C1F7T523G0.md",
  "msc": [
    "35J25",
    "35J40"
  ],
  "name": "Elliptic boundary value problem",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/C1F/C1F7T523G0.md",
  "status": null,
  "tag": "C1F7T523G0",
  "uri": "m2://C1F7T523G0",
  "url": "https://tamnd.github.io/math2/wiki/elliptic-boundary-value-problem/",
  "writing": {
    "tag": "C1F7T523G0",
    "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
      }
    ]
  }
}