We extend results proved by the second author (Amer. J. Math., 2009) for nonnegatively curved Alexandrov spaces to general compact Alexandrov spaces $X$ with curvature bounded below. The gradient flow of a geodesically convex functional on the quadratic Wasserstein space $\left( \mathcal{P}\left( X \right),\,{{W}_{2}} \right)$ satisfies the evolution variational inequality. Moreover, the gradient flow enjoys uniqueness and contractivity. These results are obtained by proving a first variation formula for the Wasserstein distance.

We prove that Alexandrov spaces of non-negative curvature have Markov type 2 in the sense of Ball. As a corollary, any Lipschitz continuous map from a subset of an Alexandrov space of non-negative curvature into a 2-uniformly convex Banach space can be extended to a Lipschitz continuous map on the entire space.

Microstructures of epitaxial Ca0.33CoO2 thin films, which were grown on m plane and c(0001) plane of α–Al2O3 by the reactive solid-phase epitaxy (R-SPE) method and the subsequent ion-exchange treatment, were investigated in detail by using selected-area electron diffraction, high-resolution transmission electron microcopy, spherical-aberration-corrected high-angle annular dark-field scanning transmission electron microscopy (Cs-corrected HAADF-STEM), and electron energy-loss spectroscopy (EELS). Detailed electron diffraction analyses reveal that the orientation relationships between Ca0.33CoO2 thin film and substrate are and , having an angle of about 43° with for the film deposited on m plane, and and for the film deposited on c(0001) plane though a Ca–Al–O amorphous layer formed between them. CoO seed layer near the interface and residual Co3O4 phase inside the films were observed and identified by HAADF-STEM and EELS in both samples. Such microstructural configuration indicates that the processes of film growth during R-SPE are (i) oxidation of CoO into Co3O4 with residual CoO layer near the interface and (ii) intercalation of Na+ layer into Co3O4 to achieve the layered NaxCoO2 film while forming Na–Al–O amorphous layer at the interface.