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  "body": "\n## Summary\n\nThe Dirichlet energy of a function $u$ on a domain is $\\frac12 \\int \\abs{\\nabla u}^2 \\, dx$, half the integral of the squared length of its gradient.\nIt measures how much the function varies.\nIts critical points with fixed boundary values are the [[harmonic functions|0DEB0JMN0W]], which is the content of Dirichlet's principle.\nThe energy is the quadratic form behind the [[Laplacian|0E0B6RA8K0]], the Sobolev space $H^1$ and the theory of Dirichlet forms.\n\n## Explanations\n\n### L1 human\n\nThink of a rubber membrane whose height over the point $x$ is $u(x)$.\nFor small slopes, the stretching energy of the membrane is roughly proportional to the Dirichlet energy.\nA flat membrane has energy $0$, and a wrinkled one has large energy.\nFor example, $u(x, y) = x$ on the unit disc has gradient of length $1$ everywhere, so its energy is $\\frac12$ times the area $\\pi$, that is, $\\pi/2$.\n",
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