  Our purpose is to obtain gradient estimates for certain nonlinear partial
differential equations by coupling methods. First we derive uniform gradient
estimates for a certain semi-linear PDEs based on the coupling method
introduced in Wang (2011) and the theory of backward SDEs. Then we generalize
Wang's coupling to the $G$-expectation space and obtain gradient estimates for
nonlinear diffusion semigroups, which correspond to the solutions of a certain
fully nonlinear PDEs.
