Some Remarks on Korn Inequalities

A recent joint paper with Doina Cioranescu and Julia Orlik was concerned with the homogenization of a linearized elasticity problem with inclusions and cracks (see [Cioranescu, D., Damlamian, A. and Orlik, J., Homogenization via unfolding in periodic elasticity Willi contact on closed and open cracks, Asymptotic Analysi,s, 82, 2013, 201232]). It required uniform estimates with respect to the homogenization parameter. A Korn inequality was used which involves unilateral terms on the boundaries where a no penetration condition is imposed. In this paper, the author presents a general method to obtain many diverse Korn inequalities including the unilateral inequalities used in [Cioranescu, D., Damlamian, A. and Orlik, J., Homogenization via unfolding in periodic elasticity with contact on closed and open cracks, Asymptotic Analysis, 82, 2013, 201-232]. A preliminary version was presented in [Damlamian, A., Some unilateral Korn inequalities with application to a contact problem with inclusions, C. R. Acad. Sci. Paris, Sea. 350, 2012, 861 8651.