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