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On The Sharpness of a Korn’s Inequality For Piecewise \(H^1\) Space and Its Applications

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Abstract

In this paper, we investigate the sharpness of a Korn’s inequality for piecewise \(H^1\) space and its applications. We first revisit a Korn’s inequality for the piecewise \(H^1\) space based on general polygonal or polyhedral decompositions of the domain. We express the Korn’s inequality with minimal jump terms. Then we prove that such minimal jump conditions are sharp for achieving the Korn’s inequality. The sharpness of the Korn’s inequality and explicitly given minimal conditions can be used to test whether any given finite element spaces satisfy Korn’s inequality, immediately as well as to build or modify nonconforming finite elements for Korn’s inequality to hold.

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Acknowledgements

The author Qingguo Hong acknowledges the support from NSF Grant NSF DMS-2419033. The author Young-Ju Lee acknowledges the partial support of Shapiro Fellowship from Penn State in Spring of 2022 and the partial support from NSF DMS-2208499.

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The author Qingguo Hong is supported by NSF Grant DMS-2419033.

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Qingguo Hong proposed the idea, proved the main results, and wrote the manuscript. Young-Ju Lee went through the paper and revised the writing, especially draw the pictures.

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Hong, Q., Lee, YJ. & Xu, J. On The Sharpness of a Korn’s Inequality For Piecewise \(H^1\) Space and Its Applications. J Sci Comput 102, 6 (2025). https://doi.org/10.1007/s10915-024-02724-w

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  • DOI: https://doi.org/10.1007/s10915-024-02724-w

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