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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $\\Omega \\subseteq \\R^n$ be an [[open set|02BH3MCBQB]].\nLet $u \\colon \\Omega \\to \\R$ be continuously differentiable, or more generally weakly differentiable with locally integrable gradient $\\nabla u$.\nThe *Dirichlet energy* of $u$ is\n$$E[u] = \\frac{1}{2} \\int_\\Omega \\abs{\\nabla u(x)}^2 \\, dx \\in [0, \\infty].$$\n\n## Remarks\n\nExample: on the unit square $(0, 1)^2$, $u(x, y) = x^2$ has $\\nabla u = (2x, 0)$, so $E[u] = \\frac12 \\int_0^1 4x^2 \\, dx = \\frac23$.\nNon-example of finiteness: on the interval $(0, 1)$, $u(x) = x^{1/4}$ is continuously differentiable, but $E[u] = \\frac12 \\int_0^1 \\frac{1}{16} x^{-3/2} \\, dx = \\infty$.\nThe functions in $L^2(\\Omega)$ with finite energy form the [[Sobolev space|M4CEZ5K9HP]] $H^1(\\Omega) = W^{1,2}(\\Omega)$.\nSome authors omit the factor $\\frac12$; the minimizers are the same.\n$E[u] = 0$ for a function on a connected open set if and only if $u$ is constant.\nThe energy is invariant under conformal maps in dimension $2$.\n",
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