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  "body": "\n## Summary\n\nAn elliptic operator is a linear differential operator whose principal symbol does not vanish at any nonzero covector.\nThe [[Laplacian|0E0B6RA8K0]] is the model example.\nElliptic operators have strong regularity properties: solutions of $Lu = f$ are as smooth as the data allow.\nOn compact manifolds they are Fredholm, which is the setting of index theory.\n\n## Explanations\n\n### L1 human\n\nA linear differential operator is a sum of derivatives of a function, each multiplied by a coefficient.\nThe terms of highest order decide most of its behaviour, and the principal symbol records them by replacing each derivative $\\partial_j$ by a variable $\\xi_j$.\nThe operator is elliptic when this polynomial in $\\xi$ vanishes only at $\\xi = 0$.\nFor the Laplacian $\\partial_1^2 + \\partial_2^2$ the symbol is $\\xi_1^2 + \\xi_2^2$, which is zero only at the origin.\nFor the wave operator $\\partial_t^2 - \\partial_x^2$ the symbol $\\tau^2 - \\xi^2$ vanishes on the lines $\\tau = \\pm \\xi$, so it is not elliptic.\nIndeed, waves can carry sharp edges, while solutions of Laplace's equation are smooth.\n",
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