On a decomposition of regular domains into John domains with uniform constants

TL;DR

This paper proves that any smooth, simply connected two-dimensional domain can be partitioned into John domains with uniform constants, enabling Sobolev inequalities to hold uniformly across the partition.

Contribution

It introduces a decomposition method for regular domains into John domains with uniform constants, independent of the original domain's shape.

Findings

Partition of domains into John domains with uniform constants

Uniform validity of Sobolev inequalities on the partition

Applicable to domains with smooth boundaries

Abstract

We derive a decomposition result for regular, two-dimensional domains into John domains with uniform constants. We prove that for every simply connected domain $\Omega \subset {\Bbb R}^2$ with $C^1$-boundary there is a corresponding partition $\Omega = \Omega_1 \cup \ldots \cup \Omega_N$ with $\sum_{j=1}^N \mathcal{H}^1(\partial \Omega_j \setminus \partial \Omega) \le \theta$ such that each component is a John domain with a John constant only depending on $\theta$. The result implies that many inequalities in Sobolev spaces such as Poincar\'e's or Korn's inequality hold on the partition of $\Omega$ for uniform constants, which are independent of $\Omega$.