Strain Tensors and Matching Property on Degenerated Hyperbolic Surfaces

We prove the regularity of solutions to the strain tensor equation on degenerated hyperbolic surfaces $S$ where the Gauss curvature is zero on a part of boundary. Furthermore, we obtain the density property that smooth infinitesimal isometries are dense in the $W^{2,2}({S},\R^3)$ infinitesimal isometries. Finally, the matching property is established. Those results are important tools in obtaining recovery sequences ($\Ga$-lim sup inequality) for dimensionally-reduced shell theories in elasticity.