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  "body": "\n## Summary\n\nThe $p$-Laplacian is the nonlinear operator $\\Delta_p u = \\operatorname{div}(\\abs{\\nabla u}^{p-2} \\nabla u)$, for a real number $1 \u003c p \u003c \\infty$.\nFor $p = 2$ it is the ordinary [[Laplacian|0E0B6RA8K0]].\nIt is the Euler–Lagrange operator of the energy $\\int \\abs{\\nabla u}^p \\, dx$, just as the Laplacian is for $\\int \\abs{\\nabla u}^2 \\, dx$.\nIt is the model of a quasilinear elliptic operator that degenerates where $\\nabla u = 0$ for $p \u003e 2$ and becomes singular there for $p \u003c 2$.\n\n## Explanations\n\n### L1 human\n\nThe Laplacian measures how a function differs from its averages, with all slopes treated alike.\nThe $p$-Laplacian gives more weight to steep slopes when $p \u003e 2$ and less weight when $p \u003c 2$.\nFunctions with $\\Delta_p u = 0$, called $p$-harmonic functions, minimize the total amount of $\\abs{\\nabla u}^p$ for given boundary values.\nEvery affine function $u(x) = a \\cdot x + b$ is $p$-harmonic, because its gradient is constant and so the divergence vanishes.\n",
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