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Korn’s inequality and John domains. (English) Zbl 1373.35015

Summary: Let \(\Omega \subset \mathbb {R}^n\), \(n\geq 2\), 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 \(\int _\Omega \frac{\partial u_i}{\partial x_j}-\frac{\partial u_j}{\partial x_i}\,dx=0\), \(1\leq i,j\leq n\), \[ \| D\mathbf {u}\| _{L^p(\Omega )}\leq C_K(\Omega , p)\| \epsilon (\mathbf {u})\| _{L^p(\Omega )}; \quad (K_{p}) \]
(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 \[ \| \mathbf {v}\| _{W^{1,p}(\Omega ,\mathbb {R}^n)}\leq C(\Omega , p)\| f\| _{L^p(\Omega )};\quad (DE_p) \]
(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 M. Costabel and M. Dauge [Arch. Ration. Mech. Anal. 217, No. 3, 873–898 (2015; Zbl 1329.35019)] and a question raised by E. Russ [Vietnam J. Math. 41, No. 4, 369–381 (2013; Zbl 1295.35187)]. 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.

MSC:

35A23 Inequalities applied to PDEs involving derivatives, differential and integral operators, or integrals
35F05 Linear first-order PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35J25 Boundary value problems for second-order elliptic equations
26D10 Inequalities involving derivatives and differential and integral operators
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References:

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