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Balls minimize moments of logarithmic and Newtonian equilibrium measures

Carrie Clark, Richard S. Laugesen

Abstract

The $q$-th moment ($q>0$) of electrostatic equilibrium measure is shown to be minimal for a centered ball among $3$-dimensional sets of given capacity, while among $2$-dimensional sets a centered disk is the minimizer for $0<q \leq 2$. Analogous results are developed for Newtonian capacity in higher dimensions and logarithmic capacity in $2$ dimensions. Open problems are raised for Riesz equilibrium moments.

Balls minimize moments of logarithmic and Newtonian equilibrium measures

Abstract

The -th moment () of electrostatic equilibrium measure is shown to be minimal for a centered ball among -dimensional sets of given capacity, while among -dimensional sets a centered disk is the minimizer for . Analogous results are developed for Newtonian capacity in higher dimensions and logarithmic capacity in dimensions. Open problems are raised for Riesz equilibrium moments.
Paper Structure (12 sections, 12 theorems, 90 equations, 1 figure)

This paper contains 12 sections, 12 theorems, 90 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Minimal moments of equilibrium measures
  3. Definitions of energy, capacity, and equilibrium measure
  4. Prior work on moments of equilibrium measure
  5. Literature on isoperimetric theorems for Riesz capacity
  6. Proof of \ref{['th:momentnewton']}
  7. Proof of \ref{['pr:momentsthreshold']}
  8. Moment formula in ${{\mathbb R}^{n+1}}$
  9. Integral operator $J$
  10. Nonnegativity of $Jv-Ju$
  11. Proof of \ref{['th:momentnone']}: the moment inequalities
  12. Potential theoretic background

Key Result

Theorem 1.1

Let $n \geq 2$. Suppose the compact set $K \subset {{\mathbb R}^n}$ has the same $(n-2)$-capacity as a closed ball $B \subset {{\mathbb R}^n}$ centered at the origin, $\operatorname{Cap_{\mathit{n}-2}}(K)=\operatorname{Cap_{\mathit{n}-2}}(B)>0$, and write $\mu$ and $\nu$ for the $(n-2)$-equilibrium (ii) Letting $q \to 0$, (iii) If $-(n-2) \leq q < 0$ then the inequality is reversed: (iv) (Equal

Figures (1)

  • Figure 1: The compact set $K$ in \ref{['th:momentnone']} lies in ${{\mathbb R}^n}$. For the proof, we regard $K$ as lying in the higher dimensional space ${{\mathbb R}^{n+1}}$ and proceed by integrating the (harmonic) equilibrium potential over the $n$-dimensional ball at height $z$ with radius $r$.

Theorems & Definitions (21)

  • Theorem 1.1: Moments of Newtonian equilibrium measure are minimal for the ball
  • Theorem 1.2: Moments of $(n-1)$-equilibrium measure are minimal for the ball
  • Corollary 1: Moments of planar sets
  • Conjecture 1: Moments of Riesz $p$-equilibrium measure are minimal for the ball
  • Proposition 1: Range of moments for $p$-equilibrium measures with $0<p<n-2$
  • Lemma 1: Spherical mean value of the potential near infinity
  • proof
  • Lemma 2: Difference of moments
  • proof : Proof of formula \ref{['eq:momentvu']} in the lemma
  • proof : Proof of formula \ref{['eq:momentJ']}
  • ...and 11 more