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

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 $\Omega \subset {\mathbb{R}}^2$ with $C^1$-boundary there is a corresponding partition $\Omega ={\Omega }_1\cup \dots \cup {\Omega }_N$ 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 Poincaré's or Korn's inequality hold on the partition of $\Omega$ for uniform constants, which are independent of $\Omega$.