On a decomposition of regular domains into John domains with uniform constants

Manuel Friedrich

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arXiv · 1605.00130

On a decomposition of regular domains into John domains with uniform constants

Manuel Friedrich

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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 $\Omega \subset {\Bbb R}^2$ with $C^1$-boundary there is a corresponding partition $\Omega = \Omega_1 \cup \ldots \cup \Omega_N$ with $\sum_{j=1}^N \mathcal{H}^1(\partial \Omega_j \setminus \partial \Omega) \le \theta$ such that each component is a John domain with a John constant only depending on $\theta$. The result implies that many inequalities in Sobolev spaces such as Poincar\'e's or Korn's inequality hold on the partition of $\Omega$ for uniform constants, which are independent of $\Omega$.

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Manuel Friedrich. 2016-04-30. On a decomposition of regular domains into John domains with uniform constants. https://arxiv.org/abs/1605.00130

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