# Korn’s inequality and John domains

Korn’s inequality and John domains Let $$\Omega \subset \mathbb {R}^n$$ Ω ⊂ R n , $$n\ge 2$$ n ≥ 2 , be a bounded domain satisfying the separation property. We show that the following conditions are equivalent: (i) $$\Omega$$ Ω is a John domain; (ii) for a fixed $$p\in (1,\infty )$$ p ∈ ( 1 , ∞ ) , the Korn inequality holds for each $$\mathbf {u}\in W^{1,p}(\Omega ,\mathbb {R}^n)$$ u ∈ W 1 , p ( Ω , R n ) satisfying $$\int _\Omega \frac{\partial u_i}{\partial x_j}-\frac{\partial u_j}{\partial x_i}\,dx=0$$ ∫ Ω ∂ u i ∂ x j - ∂ u j ∂ x i d x = 0 , $$1\le i,j\le n$$ 1 ≤ i , j ≤ 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} ‖ D u ‖ L p ( Ω ) ≤ C K ( Ω , p ) ‖ ϵ ( u ) ‖ L p ( Ω ) ; ( K p ) (ii’) for all $$p\in (1,\infty )$$ p ∈ ( 1 , ∞ ) , $$(K_p)$$ ( K p ) holds on $$\Omega$$ Ω ; (iii) for a fixed $$p\in (1,\infty )$$ p ∈ ( 1 , ∞ ) , for each $$f\in L^p(\Omega )$$ f ∈ L p ( Ω ) with vanishing mean value on $$\Omega$$ Ω , there exists a solution $$\mathbf {v}\in W^{1,p}_0(\Omega ,\mathbb {R}^n)$$ v ∈ W 0 1 , p ( Ω , R n ) to the equation $$\mathrm {div}\,\mathbf {v}=f$$ div 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} ‖ v ‖ W 1 , p ( Ω , R n ) ≤ C ( Ω , p ) ‖ f ‖ L p ( Ω ) ; ( D E p ) (iii’) for all $$p\in (1,\infty )$$ p ∈ ( 1 , ∞ ) , $$(DE_p)$$ ( D E p ) holds on $$\Omega$$ Ω . 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Calculus of Variations and Partial Differential Equations Springer Journals

# Korn’s inequality and John domains

, Volume 56 (4) – Jul 10, 2017
18 pages

/lp/springer_journal/korn-s-inequality-and-john-domains-DTYQKe9vn4
Publisher
Springer Berlin Heidelberg
Subject
Mathematics; Analysis; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization; Theoretical, Mathematical and Computational Physics
ISSN
0944-2669
eISSN
1432-0835
D.O.I.
10.1007/s00526-017-1196-7
Publisher site
See Article on Publisher Site

### Abstract

Let $$\Omega \subset \mathbb {R}^n$$ Ω ⊂ R n , $$n\ge 2$$ n ≥ 2 , be a bounded domain satisfying the separation property. We show that the following conditions are equivalent: (i) $$\Omega$$ Ω is a John domain; (ii) for a fixed $$p\in (1,\infty )$$ p ∈ ( 1 , ∞ ) , the Korn inequality holds for each $$\mathbf {u}\in W^{1,p}(\Omega ,\mathbb {R}^n)$$ u ∈ W 1 , p ( Ω , R n ) satisfying $$\int _\Omega \frac{\partial u_i}{\partial x_j}-\frac{\partial u_j}{\partial x_i}\,dx=0$$ ∫ Ω ∂ u i ∂ x j - ∂ u j ∂ x i d x = 0 , $$1\le i,j\le n$$ 1 ≤ i , j ≤ 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} ‖ D u ‖ L p ( Ω ) ≤ C K ( Ω , p ) ‖ ϵ ( u ) ‖ L p ( Ω ) ; ( K p ) (ii’) for all $$p\in (1,\infty )$$ p ∈ ( 1 , ∞ ) , $$(K_p)$$ ( K p ) holds on $$\Omega$$ Ω ; (iii) for a fixed $$p\in (1,\infty )$$ p ∈ ( 1 , ∞ ) , for each $$f\in L^p(\Omega )$$ f ∈ L p ( Ω ) with vanishing mean value on $$\Omega$$ Ω , there exists a solution $$\mathbf {v}\in W^{1,p}_0(\Omega ,\mathbb {R}^n)$$ v ∈ W 0 1 , p ( Ω , R n ) to the equation $$\mathrm {div}\,\mathbf {v}=f$$ div 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} ‖ v ‖ W 1 , p ( Ω , R n ) ≤ C ( Ω , p ) ‖ f ‖ L p ( Ω ) ; ( D E p ) (iii’) for all $$p\in (1,\infty )$$ p ∈ ( 1 , ∞ ) , $$(DE_p)$$ ( D E p ) holds on $$\Omega$$ Ω . 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.

### Journal

Calculus of Variations and Partial Differential EquationsSpringer Journals

Published: Jul 10, 2017

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