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  "body": "\n## Statement\n\nLet $n \\ge 2$, let $\\Omega \\subseteq \\R^n$ be a bounded [[open set|02BH3MCBQB]] with $C^1$ [[boundary|05D5EGMFYV]], and let $g$ be a [[continuous function|0X3WGX95FE]] on $\\partial\\Omega$.\nLet $\\mathcal{A}$ be the set of functions $w$ that are twice continuously differentiable on $\\overline{\\Omega}$ with $w = g$ on $\\partial\\Omega$.\nLet $E[w] = \\frac12 \\int_\\Omega \\abs{\\nabla w}^2 \\, dx$ be the [[Dirichlet energy|YG3W8CBB3E]].\nFor $u \\in \\mathcal{A}$, the following are equivalent:\n\n1. $u$ is [[harmonic|0DEB0JMN0W]] on $\\Omega$.\n2. $E[u] \\le E[w]$ for every $w \\in \\mathcal{A}$.\n\n## Remarks\n\nProof idea: for $w \\in \\mathcal{A}$ put $v = w - u$, which vanishes on $\\partial\\Omega$.\nThen $E[w] = E[u] + \\int_\\Omega \\nabla u \\cdot \\nabla v \\, dx + E[v]$, and integration by parts gives $\\int_\\Omega \\nabla u \\cdot \\nabla v \\, dx = -\\int_\\Omega v \\, \\Delta u \\, dx$.\nIf $u$ is harmonic, the middle term is $0$, so $E[w] \\ge E[u]$.\nIf $u$ minimizes, the derivative of $t \\mapsto E[u + tv]$ at $t = 0$ is $0$ for all such $v$, which forces $\\Delta u = 0$.\nThe theorem does not say that a minimizer exists in $\\mathcal{A}$; Weierstrass pointed out that this needs proof, and it was later justified by Hilbert.\nExistence can fail even for continuous $g$: on the unit disc, $g(\\theta) = \\sum_{k \\ge 1} k^{-2} \\cos(2^k \\theta)$ is continuous, but its harmonic extension $\\sum_k k^{-2} r^{2^k} \\cos(2^k\\theta)$ has energy $\\frac{\\pi}{2}\\sum_k 2^k k^{-4} = \\infty$.\n\n## Explanations\n\n### L1 human\n\nAmong all ways to fill in a function with given boundary values, the harmonic one is the least wiggly.\nIt has the smallest Dirichlet energy.\nThis is why a soap film or a stretched membrane with a fixed rim takes the shape of a harmonic function, for small slopes.\n\n### L2 human\n\nDirichlet's principle turns the [[Dirichlet problem|8D56GQ4AG8]] into a minimization problem.\nIn modern form, one minimizes the energy over the Sobolev space $H^1$ with given trace.\nThe minimizer exists by the Hilbert space projection theorem, and it is a weak solution of Laplace's equation.\nElliptic regularity then shows that the weak solution is a classical harmonic function.\nThe same variational idea gives existence for many other elliptic equations.\n",
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