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  "body": "\n## Problem\n\nLet $E \\subseteq \\C$ be a closed set.\nLet $w \\colon E \\to [0, \\infty)$ be an upper semicontinuous weight such that the set $\\set{z \\in E : w(z) \u003e 0}$ has positive [[logarithmic capacity|DBQXRTC0VN]].\nIf $E$ is unbounded, assume also that $\\abs{z}\\, w(z) \\to 0$ as $\\abs{z} \\to \\infty$ in $E$.\nLet $Q = -\\log w$ be the *external field*.\nFor a Borel [[probability measure|059TJN0NZ1]] $\\mu$ with support in $E$, set\n$$I_w(\\mu) = \\int \\int \\log \\frac{1}{\\abs{z - t}\\, w(z)\\, w(t)}\\, d\\mu(z)\\, d\\mu(t).$$\nThe problem is to find $V_w = \\inf_\\mu I_w(\\mu)$, a measure $\\mu_w$ that attains it, and its support $S_w$.\n\n## Remarks\n\nThe weighted energy equals $I(\\mu) + 2\\int Q\\, d\\mu$, where $I(\\mu)$ is the logarithmic energy.\nSo the field $Q$ pushes the charge away from where $Q$ is large.\nFor $w = 1$ on a compact set $E$ this is the problem of the [[equilibrium measure|464NN2K4FZ]] of $E$.\nGauss considered the analogous problem for the Newtonian kernel in space.\nThe weighted logarithmic version in the plane is the subject of the book of Saff and Totik (1997).\nExistence and uniqueness are known under the hypotheses above: $V_w$ is finite, there is a unique minimiser $\\mu_w$, and $S_w$ is compact with positive capacity.\nThe minimiser is characterised by Frostman-type conditions on its [[logarithmic potential|46NHWJHW44]] $U^{\\mu_w}$.\nFor a constant $F_w$, one has $U^{\\mu_w} + Q \\ge F_w$ quasi-everywhere on $E$ and $U^{\\mu_w} + Q \\le F_w$ on $S_w$.\nFinding $S_w$ for a given weight is the hard part, and it is done explicitly only for special weights.\nExample: on $E = \\R$ with $w(x) = e^{-x^2}$, the support is $[-1, 1]$ and $\\mu_w$ has the semicircle density $\\frac{2}{\\pi}\\sqrt{1 - x^2}$.\n\n## Explanations\n\n### L1 human\n\nPut a unit of charge on a conductor in the plane, but add an outside force that pulls the charge towards some places and pushes it from others.\nThe charge settles in the arrangement of least total energy.\nThe question is what this arrangement is, and in particular which part of the conductor it covers.\nSuch arrangements describe where zeros of orthogonal polynomials and eigenvalues of large random matrices accumulate.\n",
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