{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $\\Omega \\subseteq \\R^n$ be an open set with smooth boundary $\\partial \\Omega$, with $n \\ge 2$, and let $m \\ge 1$.\nLet $A$ be a linear differential operator of order $2m$ with smooth coefficients on $\\overline{\\Omega}$, with [[principal symbol|F31T650DKY]] $a(x, \\xi)$.\nFor $x \\in \\partial \\Omega$, let $\\nu(x)$ be the inner unit normal.\nAssume that $A$ is properly [[elliptic|XY8GEQSWPR]].\nThat is, let $x \\in \\partial \\Omega$ and let $\\xi' \\in \\R^n$ be nonzero and tangent to $\\partial \\Omega$ at $x$.\nThen the polynomial $\\tau \\mapsto a(x, \\xi' + \\tau \\nu(x))$ has exactly $m$ roots with positive imaginary part.\nThe roots are counted with multiplicity.\nLet $B_1, \\ldots, B_m$ be linear differential operators on $\\partial \\Omega$ with smooth coefficients, of orders $m_j \u003c 2m$ and principal symbols $b_j(x, \\xi)$.\nWrite $D_t = -i \\, d/dt$.\nThe operators $B_1, \\ldots, B_m$ satisfy the *Lopatinskii–Shapiro condition* with respect to $A$ if the following holds.\nLet $x \\in \\partial \\Omega$ and let $\\xi'$ be a nonzero tangent vector at $x$.\nThe only bounded solution $v \\colon [0, \\infty) \\to \\C$ of\n$$a\\big(x, \\xi' + \\nu(x) D_t\\big) v(t) = 0 \\quad (t \u003e 0), \\qquad b_j\\big(x, \\xi' + \\nu(x) D_t\\big) v(0) = 0 \\quad (j = 1, \\ldots, m)$$\nis $v = 0$.\n\n## Remarks\n\nEquivalently, the polynomials $\\tau \\mapsto b_j(x, \\xi' + \\tau \\nu)$ are linearly independent modulo $\\prod_k (\\tau - \\tau_k^+)$, where $\\tau_k^+$ are the $m$ roots of $a(x, \\xi' + \\tau \\nu)$ with positive imaginary part.\nExample: $A = -\\Delta$ with the Dirichlet condition $B_1 u = u$ on the half-space $\\set{x_n \u003e 0}$.\nHere $a(\\xi) = \\abs{\\xi}^2$, $\\nu = e_n$, and the ordinary differential equation is $-v'' + \\abs{\\xi'}^2 v = 0$, whose bounded solutions are $v(t) = c \\, e^{-\\abs{\\xi'} t}$; the condition $v(0) = 0$ forces $c = 0$.\nThe Neumann condition $\\partial_\\nu u$ also passes: $v'(0) = -c \\abs{\\xi'} = 0$ forces $c = 0$.\nNon-example: in $\\R^3$, take $A = -\\Delta$ on $\\set{x_3 \u003e 0}$ with $B_1 u = \\partial_1 u$ on the boundary.\nFor $\\xi' = (0, 1, 0)$, the function $v(t) = e^{-t}$ is a bounded nonzero solution, since the symbol of $B_1$ is $i \\xi_1 = 0$, so the condition fails.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "48MEJEMEXM",
    "kind": "definition",
    "msc": [
      "35J40",
      "35J25"
    ],
    "name": "Lopatinskii–Shapiro condition",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Elliptic_boundary_value_problem"
      }
    ],
    "tag": "BGRWCAT4SZ",
    "uses": [
      "XY8GEQSWPR",
      "F31T650DKY"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/BGRWCAT4SZ/",
  "links": [
    {
      "field": "defines",
      "target": "48MEJEMEXM"
    },
    {
      "field": "uses",
      "target": "XY8GEQSWPR"
    },
    {
      "field": "uses",
      "target": "F31T650DKY"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/BGRWCAT4SZ.md",
  "msc": [
    "35J40",
    "35J25"
  ],
  "name": "Lopatinskii–Shapiro condition",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/BGR/BGRWCAT4SZ.md",
  "status": null,
  "tag": "BGRWCAT4SZ",
  "uri": "m2://BGRWCAT4SZ",
  "url": "https://tamnd.github.io/math2/wiki/lopatinskii-shapiro-condition-definition/",
  "writing": {
    "tag": "BGRWCAT4SZ",
    "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
      }
    ]
  }
}