We derive a decomposition result for regular, two-dimensional domains into John domains with uniform constants. We prove that for every simply connected domain $\Omega \subset {\mathbb{R}}^{2}$ with ${C}^{1}$-boundary there is a corresponding partition _{} $\Omega ={\Omega}_{1}\cup ...\cup \phantom{\rule{4pt}{0ex}}{\Omega}_{N}$ with ${\Sigma}_{j=1}^{N}{\mathscr{H}}^{1}(\partial \phantom{\rule{0.166667em}{0ex}}{\Omega}_{j}\setminus \partial \phantom{\rule{0.166667em}{0ex}}\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 Poincaré’s or Korn’s inequality hold on the partition of $\Omega $ for uniform constants, which are independent of $\Omega$.

Keywords: John domains, Korn’s inequality, free discontinuity problems, shape optimization problems

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@article{COCV_2018__24_4_1541_0, author = {Friedrich, Manuel}, title = {On a decomposition of regular domains into {John} domains with uniform constants}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {1541--1583}, publisher = {EDP-Sciences}, volume = {24}, number = {4}, year = {2018}, doi = {10.1051/cocv/2017029}, zbl = {1414.26032}, mrnumber = {3922431}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2017029/} }

TY - JOUR AU - Friedrich, Manuel TI - On a decomposition of regular domains into John domains with uniform constants JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2018 SP - 1541 EP - 1583 VL - 24 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2017029/ DO - 10.1051/cocv/2017029 LA - en ID - COCV_2018__24_4_1541_0 ER -

%0 Journal Article %A Friedrich, Manuel %T On a decomposition of regular domains into John domains with uniform constants %J ESAIM: Control, Optimisation and Calculus of Variations %D 2018 %P 1541-1583 %V 24 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2017029/ %R 10.1051/cocv/2017029 %G en %F COCV_2018__24_4_1541_0

Friedrich, Manuel. On a decomposition of regular domains into John domains with uniform constants. ESAIM: Control, Optimisation and Calculus of Variations, Volume 24 (2018) no. 4, pp. 1541-1583. doi : 10.1051/cocv/2017029. http://archive.numdam.org/articles/10.1051/cocv/2017029/

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