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Riesz s-Equilibrium Measures on d-Rectifiable Sets as s Approaches d

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Abstract

Let A be a compact set in \({\mathbb R}^{p}\) of Hausdorff dimension d. For s ∈ (0,d) the Riesz s-equilibrium measure μ s is the unique Borel probability measure with support in A that minimizes

$$ {I_s}(\mu):=\int\int{\frac{1}{{|{x} - {y}|}^{s}}}d\mu(y)d\mu(x) $$

over all such probability measures. If A is strongly \(({\mathcal H}^d, d\kern.5pt)\)-rectifiable, then μ s converges in the weak-star topology to normalized d-dimensional Hausdorff measure restricted to A as s approaches d from below.

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Authors and Affiliations

  1. Department of Mathematics, Vanderbilt University, Nashville, TN, 37240, USA

    Matthew T. Calef & Douglas P. Hardin

Authors
  1. Matthew T. Calef
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  2. Douglas P. Hardin
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Correspondence to Matthew T. Calef.

Additional information

This research was supported, in part, by the U. S. National Science Foundation under grants DMS-0505756 and DMS-0808093.

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Calef, M.T., Hardin, D.P. Riesz s-Equilibrium Measures on d-Rectifiable Sets as s Approaches d . Potential Anal 30, 385–401 (2009). https://doi.org/10.1007/s11118-009-9122-z

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  • DOI: https://doi.org/10.1007/s11118-009-9122-z

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