  In this paper we show that two Lagrangian graphs over the torus in
$\mathbb{C}^n$ with large Lagrangian phase can be connected via Lipschitz
continuous geodesic with respect to the $L^2$ metric on the space of Lagrangian
submanifolds. In particular, the geodesic for Lagrangian graphs over the torus
in $\mathbb{C}^n$ can be formulated as a degenerate elliptic equation, and we
construct geodesic by solving the corresponding Dirichlet problem.
