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  "body": "\n## Problem\n\nLet $n \\ge 3$, let $K \\subseteq \\R^n$ be a [[compact set|0R6EWFKRCF]] of positive [[capacity|4YZVFGJVSZ]], and let $f \\colon K \\to \\R$ be a continuous function.\nFor a positive Borel measure $\\mu$ on $K$, let $U^\\mu(x) = \\int_K \\abs{x - y}^{2-n} \\, d\\mu(y)$ be its [[Newtonian potential|DCMFQ01QGS]].\nThe *Gauss integral* of $\\mu$ is\n$$G_f(\\mu) = \\int_K \\left( U^\\mu - 2 f \\right) d\\mu,$$\nthat is, the [[energy|SFSRQCM3EG]] of $\\mu$ minus twice the integral of $f$.\nThe problem asks for a positive Borel measure $\\mu$ on $K$ with $\\mu(K) = 1$ that makes $G_f(\\mu)$ as small as possible.\n\n## Remarks\n\nUnder these hypotheses a minimizing measure exists and is unique.\nExistence follows from the weak-star compactness of the probability measures on $K$ and the lower semicontinuity of the energy.\nUniqueness follows from the strict positive definiteness of the Newtonian energy on signed measures of finite energy.\nFor $f = 0$ the problem is to minimize the energy among unit measures on $K$, and its solution is the [[equilibrium measure|464NN2K4FZ]] of $K$.\nThe minimizer $\\mu$ is characterized by two inequalities with one constant $c$: $U^\\mu - f \\ge c$ on $K$ outside a set of capacity zero, and $U^\\mu - f \\le c$ on the support of $\\mu$.\nGauss treated the problem for forces that follow the inverse square law.\nRigorous proofs of existence rely on the measure theory that was developed later.\n\n## Explanations\n\n### L1 human\n\nPut a fixed amount of charge on a conductor $K$ that also feels an outside electric field $f$.\nThe charges repel each other but are pulled by the field.\nThe Gauss problem asks where the charge ends up when the total energy is as small as possible.\nWithout an outside field the answer is the equilibrium distribution, whose total energy measures the capacity of $K$.\n",
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      "first_stated": "1840",
      "posed_by": "Carl Friedrich Gauss"
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    "kind": "problem",
    "msc": [
      "31B15",
      "31A15"
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    "name": "Gauss variational problem",
    "sources": [
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        "url": "https://en.wikipedia.org/wiki/Capacity_of_a_set"
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    "tag": "94GSMEWPTN",
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  "msc": [
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  "name": "Gauss variational problem",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/94G/94GSMEWPTN.md",
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