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  "body": "\n## Statement\n\nLet $\\Omega \\subseteq \\R^n$ be open, and let $P = \\sum_{\\abs{\\alpha} \\le m} a_\\alpha(x) \\, \\partial^\\alpha$ be a [[linear differential operator|6DZETZ5T9G]] of order $m$ whose coefficients $a_\\alpha$ are [[real-analytic|081WKEDK6C]] on $\\Omega$.\nLet $S \\subseteq \\Omega$ be a real-analytic hypersurface and let $x_0 \\in S$ be a point at which $S$ is [[noncharacteristic|CQJ2E2ARRF]] for $P$.\nLet $U \\subseteq \\Omega$ be an open neighbourhood of $x_0$, and let $u \\in C^m(U)$ satisfy $P u = 0$ in $U$ and $\\partial^\\alpha u = 0$ on $S \\cap U$ for all $\\abs{\\alpha} \\le m - 1$.\nThen $u = 0$ in some neighbourhood of $x_0$.\n\n## Remarks\n\nEquivalently, two $C^m$ solutions of $P u = f$ with the same Cauchy data on $S$ coincide near $x_0$; here $f$ need not be analytic.\nThe proof is by duality: one solves the adjoint equation $P^{*} v = p$ for polynomials $p$, with zero Cauchy data on a nearby analytic hypersurface, using the [[Cauchy–Kovalevskaya theorem|97HMV9BT2W]].\nIntegration by parts then shows that $\\int u \\, p = 0$ on a lens-shaped region for every polynomial $p$, and the Weierstrass approximation theorem gives $u = 0$ there.\nExample: for the wave equation $\\partial_t^2 u - \\partial_x^2 u = 0$, the line $t = 0$ is noncharacteristic, so a $C^2$ solution with $u(0, x) = \\partial_t u(0, x) = 0$ vanishes near the line.\nThe analyticity of the coefficients cannot simply be dropped: there are linear operators with smooth coefficients for which the Cauchy problem has nonzero solutions with zero data.\nHörmander extended the theorem to distribution solutions and to $C^1$ hypersurfaces.\n\n## Explanations\n\n### L1 human\n\nThe Cauchy–Kovalevskaya theorem gives uniqueness only among analytic solutions.\nHolmgren's theorem says that for linear equations with analytic coefficients there is no other, non-analytic, solution with the same starting data.\nSo the starting data on a good surface determine the solution near the surface.\n\n### L2 human\n\nThe theorem is a unique continuation result across noncharacteristic hypersurfaces.\nIt applies to the Laplace, heat and wave equations, since their coefficients are constant.\nFor the wave equation it is a step towards domain of dependence arguments.\nFor the Laplace equation it shows that the Cauchy problem has at most one solution, even though it is not [[well posed|W09B9NZ9B5]].\nThe noncharacteristic hypothesis matters: for the heat operator the line $t = 0$ is characteristic, and [[Tychonoff's example|RN4J24CD83]] gives a nonzero solution of the heat equation with zero initial values.\n",
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