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  "body": "\n## Statement\n\nLet $U \\subseteq \\R^n$ be an open set and let $u$ be a locally integrable function on $U$.\nAssume that\n$$\\int_U u(x) \\, \\Delta\\varphi(x) \\, dx = 0$$\nfor every infinitely differentiable function $\\varphi$ with compact support in $U$ (a [[test function|HS65DW0HXH]]).\nThen there is a [[harmonic function|0DEB0JMN0W]] $h$ on $U$ with $u = h$ almost everywhere.\n\n## Remarks\n\nThe hypothesis says that $\\Delta u = 0$ in the sense of distributions, where $\\Delta$ is the [[Laplacian|0E0B6RA8K0]].\nProof idea: let $\\eta_\\varepsilon$ be a smooth radial mollifier supported in $B(0, \\varepsilon)$.\nLet $u_\\varepsilon = u * \\eta_\\varepsilon$, a [[convolution|2Z29JN3VES]] defined at the points at distance more than $\\varepsilon$ from the complement of $U$.\nThen $\\Delta u_\\varepsilon = u * \\Delta\\eta_\\varepsilon = 0$ by the hypothesis, so $u_\\varepsilon$ is smooth and harmonic.\nBy the [[mean value property|0HYPR64JM2]], $u_\\varepsilon * \\eta_\\delta = u_\\varepsilon$, so $u_\\varepsilon = u_\\varepsilon * \\eta_\\delta = u_\\delta * \\eta_\\varepsilon = u_\\delta$ where both are defined.\nThus the $u_\\varepsilon$ fit together into one harmonic function $h$, and $u_\\varepsilon \\to u$ almost everywhere as $\\varepsilon \\to 0$ gives $u = h$ almost everywhere.\nThe same holds for distributions: a distribution $T$ on $U$ with $\\Delta T = 0$ is given by a harmonic function.\nExample: $u(x) = \\abs{x}^{2-n}$ on $\\R^n$, $n \\ge 3$, is locally integrable and harmonic off $0$.\nBut $\\Delta u$ is a nonzero multiple of the Dirac delta, so the hypothesis fails on $U = \\R^n$, and indeed $u$ has no harmonic version there.\n\n## Explanations\n\n### L1 human\n\nWeyl's lemma says that it does not matter how weakly one solves Laplace's equation.\nSuppose a function is only known to satisfy it \"on average\", tested against smooth bumps.\nThen it is a genuine smooth harmonic function after a change on a set of measure zero.\n\n### L2 human\n\nThe lemma is the starting point of elliptic regularity: weak solutions of elliptic equations are as smooth as the data allow.\nIt is used, for instance, in Hilbert space methods for the Dirichlet problem, where Dirichlet's principle produces a weak solution, and Weyl's lemma upgrades it to a classical harmonic function.\nIn complex analysis it shows that a distributional solution of the Cauchy–Riemann equations is holomorphic.\n",
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