A Limiting Case for Riesz s-Energies
by
Matthew Calef
Vanderbilt University
Coauthors: Douglas Hardin

Let A be a compact subset of R^p with Hausdorff dimension d. For 0 < s < d, let I_s(m) denote the double integral over |x-y|^-s with respect to m. It is known that there is a unique equilibrium measure, m_s, that minimizes I_s over the set M(A) of Borel probability measures supported on A. For s ≥ d, the quantity I_s is not finite for any measure m in M(A). We show that, for a class of sets, which includes compact C¹-manifolds, the normalized d-energy defined as

~ I   d  (m) : = lim s↑ d  (d-s)Is(m)

exists as an extended real number for any measure in M(A), and is minimized by normalized Hausdorff measure restricted to A denoted by l_d. Further, we show that m_s converges in the weak-star topology to l_d as s approaches d from below.

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Date received: February 12, 2008