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The highest smoothness of the Green function implies the highest density of a set. (English) Zbl 1061.31003

Let \(E\) be a subset of the interval \([0,1]\) and \(\Omega\) be the complement of \(E\) in the extended plane. The author observes that the density of \(E\) at zero is somehow related to the smoothness of the Green function \(g_\Omega\) of \(\Omega\) at the origin. Among other results, he shows that if the Green function \(g_\Omega\) satisfies a Lipschitz condition \(g_\Omega(z)\leq c|z|^{1/2}\), \(z\in C\) at the origin, then the density of \(E\) in a small neighborhood of zero, measured in terms of logarithmic capacity, is arbitrarily close to the density of \([0,1]\) in that neighborhood.

MSC:

31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
35A08 Fundamental solutions to PDEs
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References:

[1] Andrievskii, V. V., On the Green function for a complement of a finite number of real intervals, to appear inConst. Approx. · Zbl 1066.30004
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[4] Ransford, T.,Potential Theory in the Complex Plane. London Mathematical Society Student Texts28, Cambridge Univ. Press, Cambridge, 1995. · Zbl 0828.31001
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[6] Totik, V., Metric Properties of Harmonic Measure,Manuscript, 2004.
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