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  "body": "\n## Definition\n\nLet $U \\subseteq \\R^n$ be an [[open set|02BH3MCBQB]] and let $f \\colon U \\to \\R$ be a given function.\n*Poisson's equation* on $U$ with right-hand side $f$ is the equation\n$$\\Delta u = f$$\nfor an unknown function $u \\colon U \\to \\R$, where $\\Delta$ is the [[Laplacian|0E0B6RA8K0]].\nA *classical solution* is a twice continuously differentiable function $u$ on $U$ with $\\Delta u(x) = f(x)$ for every $x \\in U$.\n\n## Remarks\n\nExample: on $U = \\R^n$ with $f = 1$, the function $u(x) = \\abs{x}^2 / (2n)$ is a classical solution, since each term $\\partial_i^2 (x_i^2 / (2n))$ equals $1/n$.\nNon-example: $u(x) = \\abs{x}^2 / 2$ on $\\R^2$ is not a solution for $f = 1$, since $\\Delta u = 2$.\nWith $f = 0$ the equation is [[Laplace's equation|S8RG6N1PRT]].\nIf $u_1$ and $u_2$ solve the same Poisson equation, then $u_1 - u_2$ is [[harmonic|0DEB0JMN0W]].\nMany texts write the equation as $-\\Delta u = f$, the sign that makes the operator positive.\nIf $f$ is twice continuously differentiable with compact support in $\\R^n$, $n \\ge 2$, then $u = \\Phi * f$ solves $-\\Delta u = f$ on $\\R^n$, where $\\Phi$ is the [[fundamental solution of the Laplacian|BSJ8X2XRN9]].\nWeak and distributional solutions are defined for less regular $f$.\n",
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