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  "body": "\n## Statement\n\nLet $n \\ge 2$ and $R \u003e 0$, let $B = \\set{x \\in \\R^n : \\abs{x} \u003c R}$ be the [[open ball|0YE5868HZB]], and let $S = \\partial B$ be its boundary sphere with surface measure $dS$.\nLet $\\sigma_{n-1}$ be the [[surface area|0WQE3G7B88]] of the unit sphere in $\\R^n$, and for $x \\in B$ and $y \\in S$ put\n$$P(x, y) = \\frac{R^2 - \\abs{x}^2}{\\sigma_{n-1} R \\, \\abs{x - y}^n}.$$\n\n1. If $u$ is [[continuous|0X3WGX95FE]] on $\\overline{B}$ and [[harmonic|0DEB0JMN0W]] on $B$, then $u(x) = \\int_S P(x, y) \\, u(y) \\, dS(y)$ for every $x \\in B$.\n2. If $g$ is continuous on $S$, then $u(x) = \\int_S P(x, y) \\, g(y) \\, dS(y)$ is harmonic on $B$, and $u(x) \\to g(y_0)$ as $x \\to y_0$ for every $y_0 \\in S$.\n\n## Remarks\n\nProof idea: $P(\\cdot, y)$ is harmonic in $B$, positive, and has integral $1$ over $S$.\nIt is minus the outward normal derivative, in the second variable, of the Green's function of the ball.\nPart 1 follows from Green's representation formula on smaller balls and a limit.\nPart 2 holds because $P(x, \\cdot)$ concentrates near $y_0$ as $x \\to y_0$, so the integral is an approximate identity.\nFor $n = 2$ and $R = 1$ the formula is $u(re^{i\\theta}) = \\frac{1}{2\\pi}\\int_0^{2\\pi} \\frac{1 - r^2}{1 - 2r\\cos(\\theta - t) + r^2} \\, u(e^{it}) \\, dt$, the classical [[Poisson kernel|TCYSWCTYYM]] of the disc.\nAt $x = 0$ the formula gives the mean value property: $u(0)$ is the average of $u$ over $S$.\nTogether, parts 1 and 2 solve the [[Dirichlet problem|8D56GQ4AG8]] on a ball, with a unique solution.\n\n## Explanations\n\n### L1 human\n\nIf you know a harmonic function on the surface of a ball, you know it inside.\nThe value at an inside point is a weighted average of the boundary values.\nBoundary points close to the inside point get large weights, and far points get small ones.\n\n### L2 human\n\nThe Poisson formula is the main tool for local results on harmonic functions.\nFrom it one gets [[Harnack's inequality|YNH832D50W]], gradient estimates, the real analyticity of harmonic functions, and the fact that uniform limits of harmonic functions are harmonic.\nIt also yields the converse of the mean value property.\nIt is the basic step in Perron's method, where a function is replaced by its Poisson integral on a small ball.\n",
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