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The \(L_p\) Minkowski problem for the electrostatic \(\mathfrak{p}\)-capacity. (English) Zbl 1453.31012

Summary: Existence and uniqueness of the solution to the \(L_p\) Minkowski problem for the electrostatic \(p\)-capacity are proved when \(p > 1\) and \(1 < p< n\). These results are nonlinear extensions of the very recent solution to the \(L_p\) Minkowski problem for \(p\)-capacity when \(p = 1\) and \(1 < p < n\) by A. Colesanti et al. [Adv. Math. 285, 1511–1588 (2015; Zbl 1327.31024)] and M. Akman et al. [“The Brunn-Minkowski inequality and a Minkowski problem for nonlinear capacity”, Preprint, arXiv:1709.00447], and the classical solution to the Minkowski problem for electrostatic capacity when \(p = 1\) and \(p= 2\) by D. Jerison [Acta Math. 176, No. 1, 1–47 (1996; Zbl 0880.35041)].

MSC:

31B15 Potentials and capacities, extremal length and related notions in higher dimensions
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
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References:

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