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  "body": "\n## Definition\n\nLet $\\Omega \\subseteq \\R^n$ be an open set and let $m \\ge 1$ be an integer.\nFor a multi-index $\\alpha = (\\alpha_1, \\ldots, \\alpha_n)$ of nonnegative integers, write $\\abs{\\alpha} = \\alpha_1 + \\cdots + \\alpha_n$, $\\partial^\\alpha = \\partial_1^{\\alpha_1} \\cdots \\partial_n^{\\alpha_n}$ for the [[partial derivatives|09EGC6F8PT]], and $\\xi^\\alpha = \\xi_1^{\\alpha_1} \\cdots \\xi_n^{\\alpha_n}$ for $\\xi \\in \\R^n$.\nLet\n$$L = \\sum_{\\abs{\\alpha} \\le m} a_\\alpha(x) \\partial^\\alpha$$\nbe a linear differential operator of order $m$ on $\\Omega$ with complex-valued continuous coefficients $a_\\alpha$.\nIts principal symbol is $\\sigma_L(x, \\xi) = \\sum_{\\abs{\\alpha} = m} a_\\alpha(x) \\xi^\\alpha$.\nThe operator $L$ is *elliptic* if\n$$\\sigma_L(x, \\xi) \\ne 0 \\quad \\text{for all } x \\in \\Omega \\text{ and all } \\xi \\in \\R^n \\setminus \\set{0}.$$\n\n## Remarks\n\nMany authors put a factor $i^m$ in the principal symbol; this does not change the condition.\nFor a second-order operator $L = \\sum_{j,k} a_{jk}(x)\\partial_j\\partial_k + \\sum_j b_j(x)\\partial_j + c(x)$ with real symmetric $(a_{jk})$, ellipticity means that the matrix $(a_{jk}(x))$ is positive definite or negative definite at each $x$.\nThe operator is *uniformly elliptic* if there is $\\theta \u003e 0$ with $\\sum_{j,k} a_{jk}(x)\\xi_j\\xi_k \\ge \\theta \\abs{\\xi}^2$ for all $x$ and $\\xi$.\nThe definition extends to systems, where the symbol is a matrix that must be invertible, and to operators between vector bundles on manifolds.\n\nExample: the [[Laplacian|0E0B6RA8K0]] $\\Delta = \\partial_1^2 + \\cdots + \\partial_n^2$ has symbol $\\abs{\\xi}^2$, which is nonzero for $\\xi \\ne 0$, so it is elliptic.\nThe Cauchy–Riemann operator $\\partial_x + i\\partial_y$ on $\\R^2$ has symbol $\\xi_1 + i\\xi_2$, also nonzero for $\\xi \\ne 0$.\n\nNon-example: the wave operator $\\partial_1^2 - \\partial_2^2$ on $\\R^2$ has symbol $\\xi_1^2 - \\xi_2^2$, which vanishes at $\\xi = (1, 1)$.\n",
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