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  "body": "\n## Definition\n\nLet $P = \\sum_{\\abs{\\alpha} \\le m} a_\\alpha(x) \\, \\partial^\\alpha$ be a [[linear differential operator|6DZETZ5T9G]] of order $m$ on an open set $\\Omega \\subseteq \\R^n$, with [[principal symbol|F31T650DKY]] $\\sigma_m(P)$.\nThe operator $P$ is *elliptic at* $x \\in \\Omega$ if $\\sigma_m(P)(x, \\xi) \\ne 0$ for every $\\xi \\in \\R^n \\setminus \\set{0}$.\nIt is *elliptic in* $\\Omega$ if it is elliptic at every point of $\\Omega$.\nThe equation $P u = f$ is then an *elliptic partial differential equation*.\n\nFor a second-order operator $\\sum_{i,j} a_{ij}(x) \\partial_i \\partial_j + \\sum_i b_i(x) \\partial_i + c(x)$ with real symmetric $(a_{ij})$, this means that the matrix $(a_{ij}(x))$ is positive definite or negative definite at each $x$.\nIt is *uniformly elliptic* in $\\Omega$ if there is a constant $\\theta \u003e 0$ with\n$$\\sum_{i,j=1}^{n} a_{ij}(x) \\, \\xi_i \\xi_j \\ge \\theta \\abs{\\xi}^2 \\quad \\text{for all } x \\in \\Omega, \\ \\xi \\in \\R^n.$$\n\n## Remarks\n\nExample: the [[Laplacian|0E0B6RA8K0]] $\\Delta = \\sum_i \\partial_i^2$ has principal symbol $\\abs{\\xi}^2$, which is nonzero for $\\xi \\ne 0$, so it is elliptic; it is uniformly elliptic with $\\theta = 1$.\nExample: the [[Cauchy–Riemann operator|PPSSTHTEDE]] $\\partial_x + i \\partial_y$ on $\\R^2$ has symbol $\\xi_1 + i \\xi_2$, which vanishes only at $\\xi = 0$, so it is elliptic of order $1$.\nNon-example: the wave operator $\\partial_t^2 - \\partial_x^2$ has symbol $\\tau^2 - \\xi^2$, which vanishes at $(\\tau, \\xi) = (1, 1)$, so it is not elliptic.\nIf all coefficients are real and $n \\ge 2$, an elliptic operator has even order $m$.\nIndeed $\\sigma_m(x, -\\xi) = (-1)^m \\sigma_m(x, \\xi)$, and a real continuous function that does not vanish on the connected unit sphere cannot change sign.\nFor nonlinear second-order equations $F(D^2 u, D u, u, x) = 0$, ellipticity is a monotonicity condition of $F$ in the matrix variable $D^2 u$.\n\n## Explanations\n\n### L1 human\n\nAn operator is elliptic when its top-order part does not vanish in any direction, which for second-order equations means that the coefficient matrix is definite.\n",
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