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

DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 12 million articles from more than
10,000 peer-reviewed journals.

All for just $49/month Explore the DeepDyve Library Unlimited reading Read as many articles as you need. Full articles with original layout, charts and figures. Read online, from anywhere. Stay up to date Keep up with your field with Personalized Recommendations and Follow Journals to get automatic updates. Organize your research It’s easy to organize your research with our built-in tools. Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. Monthly Plan • Read unlimited articles • Personalized recommendations • No expiration • Print 20 pages per month • 20% off on PDF purchases • Organize your research • Get updates on your journals and topic searches$49/month

14-day Free Trial

Best Deal — 39% off

Annual Plan

• All the features of the Professional Plan, but for 39% off!
• Billed annually
• No expiration
• For the normal price of 10 articles elsewhere, you get one full year of unlimited access to articles.

$588$360/year

billed annually

14-day Free Trial