Potential Analysis

, Volume 30, Issue 4, pp 385–401 | Cite as

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

  • Matthew T. Calef
  • Douglas P. Hardin


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.


Riesz potential Equilibrium measure d-rectifiable 

Mathematics Subject Classifications (2000)

Primary 31C15 Secondary 35Q99 


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Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Department of MathematicsVanderbilt UniversityNashvilleUSA

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