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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
Article

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.

Keywords

Riesz potential Equilibrium measure d-rectifiable 

Mathematics Subject Classifications (2000)

Primary 31C15 Secondary 35Q99 

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References

  1. 1.
    Bedford, T., Fisher, A.M.: Analogues of the Lebesgue density theorem for fractal sets of reals and integers. Proc. London Math. Soc. (3) 64(1), 95–124 (1992)CrossRefMathSciNetMATHGoogle Scholar
  2. 2.
    Borodachov, S., Hardin, D., Saff, E.: Asymptotics for discrete weighted minimal energy problems on rectifiable sets. Trans. Amer. Math. Soc. 360(3), 1559–1580 (2008)CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    Federer, H.: Geometric Measure Theory, 1st edn. Springer, New York (1969)MATHGoogle Scholar
  4. 4.
    Götz, M.: On the Riesz energy of measures. J. Approx. Theory 122(1), 62–78 (2003)CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    Hardin, D., Saff, E.: Minimal riesz energy point configurations for rectifiable d-dimensional manifolds. Adv. Math. 193, 174–204 (2005)CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Hardin, D.P., Saff, E.B.: Discretizing manifolds via minimum energy points. Notices Amer. Math. Soc. 51(10), 1186–1194 (2004)MathSciNetMATHGoogle Scholar
  7. 7.
    Hinz, M.: Average densities and limits of potentials. Master’s thesis, Universität Jena, Jena (2005)Google Scholar
  8. 8.
    Landkof, N.S.: Foundations of Modern Potential Theory. Springer, New York (1973)Google Scholar
  9. 9.
    Mattila, P.: Geometry of Sets and Measures in Euclidian Spaces. Cambridge University Press, Cambridge (1995)Google Scholar
  10. 10.
    Putinar, M.: A renormalized Riesz potential and applications. In: Advances in Constructive Approximation: Vanderbilt 2003, Mod. Methods Math., pp. 433–465. Nashboro, Brentwood (2004)Google Scholar
  11. 11.
    Wolff, T.H.: Lectures on Harmonic Analysis, University Lecture Series, vol. 29. American Mathematical Society, Providence (2003)Google Scholar
  12. 12.
    Zähle, M.: The average density of self-conformal measures. J. London Math. Soc. (2) 63(3), 721–734 (2001)CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Department of MathematicsVanderbilt UniversityNashvilleUSA

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