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Weighted asymptotic Korn and interpolation Korn inequalities with singular weights


Authors: Davit Harutyunyan and Hayk Mikayelyan
Journal: Proc. Amer. Math. Soc. 147 (2019), 3635-3647
MSC (2010): Primary 00A69, 35J65, 74B05, 74B20, 74K25
DOI: https://doi.org/10.1090/proc/14533
Published electronically: May 9, 2019
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Abstract: In this work we derive asymptotically sharp weighted Korn and Korn-like interpolation (or first and a half) inequalities in thin domains with singular weights. The constants $ K$ (Korn's constant) in the inequalities depend on the domain thickness $ h$ according to a power rule $ K=Ch^\alpha ,$ where $ C>0$ and $ \alpha \in R$ are constants independent of $ h$ and the displacement field. The sharpness of the estimates is understood in the sense that the asymptotics $ h^\alpha $ is optimal as $ h\to 0.$ The choice of the weights is motivated by several factors; in particular a spatial case occurs when making Cartesian to polar change of variables in two dimensions.


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Additional Information

Davit Harutyunyan
Affiliation: Department of Mathematics, University of California Santa Barbara, Santa Barbara, California 93106
Email: harutyunyan@ucsb.edu

Hayk Mikayelyan
Affiliation: Department of Mathematical Sciences, University of Nottingham, Ningbo, 315100 People’s Republic of China
Email: Hayk.Mikayelyan@nottingham.edu.cn

DOI: https://doi.org/10.1090/proc/14533
Keywords: Korn inequality, weighted Korn inequality, thin domains
Received by editor(s): October 6, 2017
Received by editor(s) in revised form: October 18, 2018
Published electronically: May 9, 2019
Communicated by: Catherine Sulem
Article copyright: © Copyright 2019 American Mathematical Society