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⊂R2 with C1-boundary there is a corresponding
partition Ω=Ω1∪…∪ΩN with ∑j=1NH1(∂Ωj∖∂Ω)≤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
Poincar\'e's or Korn's inequality hold on the partition of Ω for uniform
constants, which are independent of Ω