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Lipschitz decompositions of domains with bilaterally flat boundaries

Jared Krandel

Abstract

We study classes of domains in $\mathbb{R}^{d+1},\ d \geq 2$ with sufficiently flat boundaries that admit a decomposition or covering of bounded overlap by Lipschitz graph domains with controlled total surface area. This study is motivated by the following result proved by Peter Jones as a piece of his proof of the Analyst's Traveling Salesman Theorem in the complex plane: Any simply connected domain $Ω\subseteq\mathbb{C}$ with finite boundary length $\mathcal{H}^1(\partialΩ)$ can be decomposed into Lipschitz graph domains with total boundary length bounded above by $M\mathcal{H}^1(\partialΩ)$ for some $M$ independent of $Ω$. In this paper, we prove an analogous Lipschitz decomposition result in higher dimensions for domains with Reifenberg flat boundaries satisfying a uniform beta-squared sum bound. We use similar techniques to show that domains with general Reifenberg flat or uniformly rectifiable boundaries admit similar Lipschitz decompositions while allowing the constituent domains to have bounded overlaps rather than be disjoint.

Lipschitz decompositions of domains with bilaterally flat boundaries

Abstract

We study classes of domains in with sufficiently flat boundaries that admit a decomposition or covering of bounded overlap by Lipschitz graph domains with controlled total surface area. This study is motivated by the following result proved by Peter Jones as a piece of his proof of the Analyst's Traveling Salesman Theorem in the complex plane: Any simply connected domain with finite boundary length can be decomposed into Lipschitz graph domains with total boundary length bounded above by for some independent of . In this paper, we prove an analogous Lipschitz decomposition result in higher dimensions for domains with Reifenberg flat boundaries satisfying a uniform beta-squared sum bound. We use similar techniques to show that domains with general Reifenberg flat or uniformly rectifiable boundaries admit similar Lipschitz decompositions while allowing the constituent domains to have bounded overlaps rather than be disjoint.
Paper Structure (28 sections, 63 theorems, 349 equations, 2 figures)

This paper contains 28 sections, 63 theorems, 349 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. Outlines of the paper and of the proofs of the theorems
  4. Preliminaries
  5. Conventions and basic definitions
  6. Reifenberg parameterizations
  7. Coherent Collections of Balls and Planes (CCBP)
  8. The definition of $g$
  9. Coronizations for Reifenberg flat and uniformly rectifiable sets
  10. Preliminaries on Lipschitz graph domains
  11. Whitney cubes and stopping time domains
  12. Carving up stopping time domains
  13. Images of Lipschitz graph domains
  14. Controlling the change in the derivative of Reifenberg parameterizations
  15. Preliminary derivative estimates and regularity
  16. ...and 13 more sections

Key Result

Theorem 1.3

There exists a constant $M>0$ such that the following holds: For any simply connected domain $\Omega\subseteq \mathbb{C}$ with $\mathcal{H}^1(\partial\Omega) < \infty$, there is a rectifiable curve $\Gamma$ such that where $\Omega_j$ is an $M$-Lipschitz graph domain for each $j$, $\Omega_j\cap\Omega_{j'}=\varnothing$ for $j\not= j'$, and

Figures (2)

  • Figure 1: A representation of $W$, $\mathop\mathrm{Cover}(W),\ \text{and } \mathop\mathrm{Divider}(W)$ in $\mathbb{R}^2$.
  • Figure 2: A representation of $[-3,3]^2\times\{0\}$ split into $\mathscr{Q}_1$ in yellow, $\mathscr{Q}_2$ in red, and $\cup_{n=3}^\infty\mathscr{Q}_n$ left uncolored at the edge of $\mathscr{Q}_2$ (The white square in the middle sits below the cube $W\in m(T)$, hence nothing above it lies in $\mathcal{D}_T$). The set $\mathop\mathrm{Divider}(W)$ shoots out of the page as a union of extensions of the sides of the squares up to the points at which they hit the slanting top of $\mathop\mathrm{Cover}(W)$.

Theorems & Definitions (135)

  • Definition 1.1: Lipschitz domains
  • Definition 1.2: Lipschitz graph domains
  • Theorem 1.3: Jon90 Theorem 2
  • Definition 1.4: Jones beta number
  • Theorem 1.5: cf. Jon90 Theorem 1, Ok92 in $\mathbb{R}^n,\ n > 2$
  • Definition 1.7: bilateral beta number
  • Definition 1.8: $(\epsilon,d)$-Reifenberg flatness
  • Theorem 1.9: cf. DT12 Theorem 1.10
  • Theorem A
  • Theorem B
  • ...and 125 more