. Korn and Korn - like inequalities for perfect cylindrical shells by Davit Harutyunyan

It is well known that Korn's inequality plays a central role in the theory of linear elasticity. In their recent work Grabovsky and Truskinovsky (2007) proved a connection between the Korn inequalities and buckling problems coming from nonlinear elasticity. It has been shown that when the general buckling theory by Grabovsky and Truskinovsky is applicable, then Korn's inequality gives a lower bound (and in some cases even the scaling) on the critical buckling load of a slender structure. In the present work we prove the asymptotically sharp Korn and Korn-like inequalities in thin rectangles in 2D with fixed and periodic Dirichlet type boundary conditions. Then, using these results we prove the asymptotically sharp Korn inequalities on the displacement gradient components. We then utilize these new results to study the buckling of compressed cylindrical shells. In particular, we calculate the critical buckling load. This is joint work with Yury Grabovsky.