  In this paper we propose new insights and ideas to set up quantitative
boundary estimates for solutions to Dirichlet problem of a class of fully
non-linear elliptic equations on compact Hermitian manifolds with real analytic
Levi flat boundary. With the quantitative boundary estimates at hand, we can
establish the gradient estimate and give a unified approach to investigate the
existence and regularity of solutions of Dirichlet problem with sufficiently
smooth boundary data, which include the geodesic equation in the space of
K\"ahler metrics as a special case. Our method can also be applied to Dirichlet
problem for analogous fully non-linear elliptic equations on a compact
Riemannian manifold with concave boundary.
