  In this paper we study elliptic PDEs on compact Gromov-Hausdorff limit spaces
of Riemannian manifolds with lower Ricci curvature bounds. In particular we
establish continuities of geometric quantities, which include solutions of
Poisson's equations, eigenvalues of Schrodinger operators, generalized Yamabe
constants and eigenvalues of the Hodge Laplacian, with respect to the
Gromov-Hausdorff topology. We apply these to the study of second-order
differential calculus on such limit spaces.
