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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
Σj=1NH1(∂Ωj\∂Ω)≤θ
Σj=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 Poincaré’s or Korn’s inequality hold on the partition of Ω for uniform constants, which are independent of Ω
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