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
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Publisher
Springer Berlin Heidelberg
Copyright
Copyright © 2017 by Springer-Verlag GmbH Germany
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

References

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