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Sharp capacity estimates in s-John domains

Chang-Yu Guo

Abstract

It is well-known that several problems related to analysis on $s$-John domains can be unified by certain capacity lower estimates. In this paper, we obtain general lower bounds of $p$-capacity of a compact set $E$ and the central Whitney cube $Q_0$ in terms of the Hausdorff $q$-content of $E$ in an $s$-John domain $Ω$. Moreover, we construct several examples to show the essential sharpness of our estimates.

Sharp capacity estimates in s-John domains

Abstract

It is well-known that several problems related to analysis on -John domains can be unified by certain capacity lower estimates. In this paper, we obtain general lower bounds of -capacity of a compact set and the central Whitney cube in terms of the Hausdorff -content of in an -John domain . Moreover, we construct several examples to show the essential sharpness of our estimates.

Paper Structure

This paper contains 4 sections, 3 theorems, 47 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Main proofs
  4. Examples

Key Result

Theorem 1.1

Let $\Omega\subset\mathbb{R}^n$, $n\geq 2$, be an $s$-John domain. For $0<\varepsilon<1$, $1\leq p\leq n$ and $q\geq s(n-1)+1-p+\varepsilon$, there exists a positive constant $C(n,p,q,s,\varepsilon)$ such that whenever $E\subset\Omega$ is a compact set disjoint from $Q_0$.

Figures (3)

  • Figure 1: The standard "room and corridors" type domain
  • Figure 2: "room and $s$-passage" type replacement
  • Figure 3: The $s$-John domain $\Omega\subset\mathbb{R}^2$

Theorems & Definitions (8)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof : Proof of Theorem \ref{['thm:main theorem']}
  • Example 4.1
  • Example 4.2
  • Example 4.3