On a decomposition of regular domains into John domains with uniform constants
Manuel Friedrich · ESAIM Control Optimisation and Calculus of Variations · 2017
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 ⊂ ℝ 2 with C 1 -boundary there is a corresponding partition Ω = Ω 1 ⋃ … ⋃ Ω N with Σ j=1 N H 1 ( ∂Ω j \ ∂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 .