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  "body": "\n## Summary\n\nAn elliptic partial differential equation is one whose [[principal symbol|F31T650DKY]] vanishes for no nonzero covector.\nFor second-order equations this means that the matrix of second-order coefficients is definite.\nElliptic equations model equilibrium states, such as steady temperatures, electrostatic potentials and membranes at rest.\nTheir solutions are as smooth as the data allow, and boundary value problems are the natural problems for them.\nSee the [[definition|3BYHP9E9VF]].\n\n## Explanations\n\n### L1 human\n\nThe model elliptic equation is the [[Laplace equation|S8RG6N1PRT]] $\\partial_x^2 u + \\partial_y^2 u = 0$.\nIts solutions, the [[harmonic functions|0DEB0JMN0W]], have the property that the value at a point is the average of the values on any circle around it.\nSo an elliptic equation averages things out: there are no preferred directions, no time, and no travelling waves.\nA soap film stretched over a wire frame is a picture of a solution: its shape inside is determined by the frame.\n",
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