Title: From buckling to rigidity of shells: Recent mathematical progress

 

In has been known that the rigidity of a shell (for instance under compression) is closely related to the optimal Korn's constant in the nonlinear Korn's first inequality (geometric rigidity estimate) for  $H^1$ fields under the appropriate conditions (with no or with Dirichlet type boundary conditions arising from the nature of the compression). In their celebrated work, Frisecke, James and Mueller (2002, 2006) derived an asymptotically sharp nonlinear geometric rigidity estimate for plates, which gave rise to a derivation of a hierarchy of nonlinear plate theories for different scaling regimes of the elastic energy depending on the thickness $h$ of the plate (the optimal constant scales like h^2). Frisecke-James-Mueller type theories have been derived by Gamma-convergence and rely on L^p compactness arguments and of course the underlying nonlinear Korn's inequality. While plate deformations have been understood almost completely, the rigidity, in particular the buckling of shells is less well understood. This is first of all due to the luck of sharp rigidity estimates for shells. In our recent work we derive linear sharp geometric estimates for shells by classifying them according to the Gaussian curvature. It turns out, that for zero Gaussian curvature (when one principal curvature is zero, the other one never vanishes) the amount of rigidity is $h^{3/2}$, for negative curvature it is $h^{4/3}$ and for positive curvature it is $h$. This results break through both in the shell buckling and nonlinear shell theories. All three estimates have completely new optimal constant scaling in any sharp geometric rigidity estimates aver appeared, and have classical flavor. This is partially joint work with Yury Grabovsky (Temple University)