Korn’s Inequalities and Their Applications in Continuum Mechanics

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Abstract
Korn’s inequalities have played a central role in the development of linear elasticity, not only in connection with the basic theoretical issues such as existence and uniqueness, but also in a variety of applications. The Korn inequalities, and other related inequalities for integrals of quadratic functionals, also arise in the analysis of viscous incompressible fluid flow. The dimensionless optimal constants appearing in these inequalities, the Korn constants, depend only on the shape of the domains of concern. Information on the geometric dependence of these constants is essential in applications. In this review article, we summarize the major results on Korn’s inequalities for bounded domains in two and three dimensions, with emphasis on results concerning the Korn constants. Some applications in continuum mechanics are also described.

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