{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "7T2RGR9XTZ"
    },
    {
      "field": "specializes",
      "target": "8A8FBH4C7Y"
    }
  ],
  "body": "\n## Summary\n\nRegularity theory is the part of the theory of partial differential equations that shows that solutions are smoother than first assumed.\nTypically one first obtains a [[weak solution|SJZX81A783]] in a [[Sobolev space|M4CEZ5K9HP]].\nThen one proves that it has more derivatives, in Sobolev or [[Hölder spaces|P101AX1ZSC]], when the coefficients and data are smooth.\nFor elliptic equations the classical results are [[elliptic regularity|2PVQY9PCT4]], the [[Schauder estimates|HSEKZMAH1V]], the [[Calderón–Zygmund estimates|RYQJH4RYB4]] and the [[De Giorgi–Nash–Moser theorem|32C3D8FG9V]].\nAn important source of the subject is [[Hilbert's nineteenth problem|QPMM2ZPWBW]] on the analyticity of minimisers.\n\n## Explanations\n\n### L1 human\n\nExistence of a solution is often easiest to prove in a weak sense, where the function may not even have classical derivatives.\nRegularity theory then asks how nice this solution really is.\nFor the Laplace equation $\\Delta u = 0$ the answer is striking: any locally integrable weak solution is smooth, and even real analytic.\nIn general, an elliptic operator $L$ of order $m$ with smooth coefficients gains $m$ derivatives.\nIf $Lu$ has $k$ derivatives in $L^2$, then $u$ has $k + m$ derivatives in $L^2$ inside the domain.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "35B65",
      "35J15",
      "35D30"
    ],
    "name": "Regularity theory",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Elliptic_operator"
      }
    ],
    "tag": "N383CX7WKS"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/N383CX7WKS/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/N383CX7WKS.md",
  "msc": [
    "35B65",
    "35J15",
    "35D30"
  ],
  "name": "Regularity theory",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/N38/N383CX7WKS.md",
  "status": null,
  "tag": "N383CX7WKS",
  "uri": "m2://N383CX7WKS",
  "url": "https://tamnd.github.io/math2/wiki/regularity-theory/",
  "writing": {
    "tag": "N383CX7WKS",
    "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
      }
    ]
  }
}