Uniform domains, John domains and quasi-isotropic domains

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 ⊂ R 2 with C 1-boundary there is a corresponding partition Ω = Ω 1 ∪. .. ∪ Ω N with N j=1 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.