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 $Ω\subset {\Bbb R}^2$ with $C^1$-boundary there is a corresponding partition $Ω= Ω_1 \cup \ldots \cup Ω_N$ with $\sum_{j=1}^N \mathcal{H}^1(\partial Ω_j \setminus \partial Ω) \le θ$ such that each component is a John domain with a John constant only depending on $θ$. The result implies that many inequalities in Sobolev spaces such as Poincaré's or Korn's inequality hold on the partition of $Ω$ for uniform constants, which are independent of $Ω$.