Abstract
Let \(\Omega \subset \mathbb {R}^n\), \(n\ge 2\), be a bounded domain satisfying the separation property. We show that the following conditions are equivalent:
-
(i)
\(\Omega \) is a John domain;
-
(ii)
for a fixed \(p\in (1,\infty )\), the Korn inequality holds for each \(\mathbf {u}\in W^{1,p}(\Omega ,\mathbb {R}^n)\) satisfying \(\int _\Omega \frac{\partial u_i}{\partial x_j}-\frac{\partial u_j}{\partial x_i}\,dx=0\), \(1\le i,j\le n\),
$$\begin{aligned} \Vert D\mathbf {u}\Vert _{L^p(\Omega )}\le C_K(\Omega , p)\Vert \epsilon (\mathbf {u})\Vert _{L^p(\Omega )}; \qquad (K_{p}) \end{aligned}$$ -
(ii’)
for all \(p\in (1,\infty )\), \((K_p)\) holds on \(\Omega \);
-
(iii)
for a fixed \(p\in (1,\infty )\), for each \(f\in L^p(\Omega )\) with vanishing mean value on \(\Omega \), there exists a solution \(\mathbf {v}\in W^{1,p}_0(\Omega ,\mathbb {R}^n)\) to the equation \(\mathrm {div}\,\mathbf {v}=f\) with
$$\begin{aligned} \Vert \mathbf {v}\Vert _{W^{1,p}(\Omega ,\mathbb {R}^n)}\le C(\Omega , p)\Vert f\Vert _{L^p(\Omega )};\qquad (DE_p) \end{aligned}$$ -
(iii’)
for all \(p\in (1,\infty )\), \((DE_p)\) holds on \(\Omega \).
For domains satisfying the separation property, in particular, for finitely connected domains in the plane, our result provides a geometric characterization of the Korn inequality, and gives positive answers to a question raised by Costabel and Dauge (Arch Ration Mech Anal 217(3):873–898, 2015) and a question raised by Russ (Vietnam J Math 41:369–381, 2013). For the plane, our result is best possible in the sense that, there exist infinitely connected domains which are not John but support Korn’s inequality.
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Acknowledgements
The authors would like to thank Prof. Pekka Koskela for suggesting this topic and for valuable discussions. R. Jiang was partially supported by National Natural Science Foundation of China (Nos. 11671039, 11626250). The research of A. Kauranen has been supported by the Academy of Finland via the Centre of Excellence in Analysis and Dynamics Research (Project No. 271983).
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Communicated by L. Ambrosio.
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Jiang, R., Kauranen, A. Korn’s inequality and John domains. Calc. Var. 56, 109 (2017). https://doi.org/10.1007/s00526-017-1196-7
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DOI: https://doi.org/10.1007/s00526-017-1196-7