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.
