KORNS INEQUALITIES AND THEIR APPLICATIONS IN CONTINUUM-MECHANICS

Kern's inequalities have played a central role in the development of l inear elasticity, not only in connection with the basic theoretical is sues such as existence and uniqueness, but also in a variety of applic ations. The Korn inequalities, and other related inequalities for inte grals of quadratic functionals, also arise in the analysis of viscous incompressible fluid flow. The dimensionless optimal constants appeari ng in these inequalities, the Kern 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 dom ains in two and three dimensions, with emphasis on results concerning the Korn constants. Some applications in continuum mechanics are also described.