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Riesz energy deformation through insulated strips

Carrie Clark, Richard S. Laugesen

Abstract

For compact sets in Euclidean space, Riesz energies whose exponents differ by $1$ are shown to arise as the endpoint cases of a one-parameter family of infinite-strip energies as the strip thickness increases from $0$ to $\infty$, under Neumann boundary conditions. An approach is suggested to a capacity conjecture of Pólya and Szegő.

Riesz energy deformation through insulated strips

Abstract

For compact sets in Euclidean space, Riesz energies whose exponents differ by are shown to arise as the endpoint cases of a one-parameter family of infinite-strip energies as the strip thickness increases from to , under Neumann boundary conditions. An approach is suggested to a capacity conjecture of Pólya and Szegő.
Paper Structure (8 sections, 11 theorems, 109 equations, 2 figures)

This paper contains 8 sections, 11 theorems, 109 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Kernel and energy definitions
  3. Results on the Neumann strip energy
  4. Strip kernel: basic properties
  5. Strip kernel: two-sided estimates
  6. Strip kernel: improved asymptotic as $t \to \infty$
  7. Proofs for the strip energy: asymptotics and $t$-derivative
  8. Computable example

Key Result

Theorem 1.1

Fix $1 \leq q < n$. If $K \subset {{\mathbb R}^n}$ is compact with finite Riesz $q$-energy then $E_K(t)$ is a $C^1$-smooth function of $t>0$ that interpolates between the $q$- and $(q-1)$-energies: where the positive constant $c_{q-1}$ is defined in cdef below. Equality holds as $t \to 0$ if $K$ is interior $(q-1)$-capacitable in ${{\mathbb R}^n}$ (as defined in sec:energyresults), meaning that i

Figures (2)

  • Figure 1: The compact set $K \subset {{\mathbb R}^n}$ in the strip $S(t)={{\mathbb R}^n}\times(-t,t)$. Points in the strip are written $\hat{x} = (x,z)$.
  • Figure 2: The heights of the points $\rho_j(\hat{y})$ are determined by repeated reflection across the strip boundaries at height $\pm t$. For example, $\rho_0(\hat{y})$ reflects to $\rho_{\pm 1}(\hat{y})$, which then reflect to $\rho_{\mp 2}(\hat{y})$, and so on. This method of reflections ensures that the kernel $G_t$ satisfies a Neumann condition at the strip boundaries.

Theorems & Definitions (21)

  • Theorem 1.1: Connecting Riesz energies for $K \subset {{\mathbb R}^n}$
  • Conjecture 1
  • Remark
  • Definition : Strip kernel by method of reflections
  • Lemma 1: Finite energy for a set in the hyperplane ${{\mathbb R}^n}$ implies unique equilibrium measure
  • proof
  • Lemma 2: Finite energy for one strip implies finiteness and continuity for all
  • proof
  • Remark
  • Theorem 3.1: Energy asymptotic as $t \to \infty$, for $K \subset {{\mathbb R}^{n+1}}$
  • ...and 11 more