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  "body": "\n## Summary\n\nLaplace's equation is the [[partial differential equation|GC5BDZ6QBC]] $\\Delta u = 0$, where $\\Delta$ is the [[Laplacian|0E0B6RA8K0]], the sum of the second partial derivatives.\nIts solutions are the [[harmonic functions|0DEB0JMN0W]].\nIt is the model elliptic equation.\nIt describes steady states: equilibrium temperatures, electrostatic and gravitational potentials in empty space, and potential flows of an ideal fluid.\nThe [[Dirichlet problem|8D56GQ4AG8]] asks for a solution with given values on the boundary of a region.\n\n## Explanations\n\n### L1 human\n\nLaplace's equation says that at every point the value of $u$ equals the average of its values nearby.\nA stretched elastic membrane fixed along a wire frame takes a shape whose height satisfies it.\nSo does the temperature in a metal plate once it has stopped changing.\nIn the plane, $u(x, y) = x^2 - y^2$ is a solution, because $u_{xx} = 2$ and $u_{yy} = -2$ add up to zero.\n",
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