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  "body": "\n## Summary\n\nA $p$-harmonic function is a solution of the [[p-Laplace equation|XJMYGY0ZZF]] $\\operatorname{div}(\\abs{\\nabla u}^{p-2} \\nabla u) = 0$, where $1 \u003c p \u003c \\infty$.\nFor $p = 2$ these are the classical [[harmonic functions|0DEB0JMN0W]].\nThey are the minimizers of the energy $\\int \\abs{\\nabla u}^p \\, dx$ with given boundary values, and they are the central objects of [[nonlinear potential theory|M923VKQVZM]].\nThe equation is nonlinear for $p \\ne 2$, so sums of $p$-harmonic functions are in general not $p$-harmonic.\n\n## Explanations\n\n### L1 human\n\nA harmonic function makes the energy $\\int \\abs{\\nabla u}^2$ as small as possible among functions with the same boundary values.\nA $p$-harmonic function does the same for the energy $\\int \\abs{\\nabla u}^p$.\nChanging the exponent changes how strongly steep slopes are penalized.\nLinear functions are $p$-harmonic for every $p$, because their gradient is constant.\nMany properties of harmonic functions survive, such as the maximum principle and a Harnack inequality, but the mean value property does not.\n",
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