ON THE RIGIDITY OF CERTAIN SURFACES WITH FOLDS AND APPLICATIONS TO SHELL THEORY

In the asymptotic theory of thin elastic shells the rigidity of the mi d-surface with kinematic boundary conditions plays an important role. Rigidity is understood in the sense of infinitesimal (linearized) rigi dity, i.e., the displacements vanish provided the variation of the fir st fundamental form vanishes. In this case the surface is also called ''stiff'', as it cannot undergo pure bendings. A stiff surface is impe rfectly stiff or perfectly stiff when the origin respectively does or does not belong to the essential spectrum of the boundary-value proble m. These questions are investigated in the framework of Douglis-Nirenb erg elliptic systems, with boundary conditions and transmission condit ions at the folds. The index properties ensures quasi-stiffness, i.e. stiffness up to a finite number of degrees of freedom. The concept of perfect stiffness is linked with estimates for the rigidity system at an appropriate level of regularity for the data and the solution. It i s proved that surfaces with folds are never perfectly stiff. It is als o shown that the transmission conditions at the folds contain more con ditions than those satisfying the Shapiro-Lopatinskii property. This l eads to certain rigidity properties of the folds. Some examples are gi ven.