Existence theory for geodesics in the space of Lagrangian submanifolds plays an important role in a program for studying the existence and uniqueness of special Lagrangians in Calai-Yau manifolds. In Euclidean space \({\mathbb {C}}^n\) , the geodesic for Lagrangian graphs can be formulated as a degenerate elliptic equation. Rubinstein and Solomon solved the corresponding Dirichlet problem of the Lagrangian geodesic equation by extending Harvey-Lawson’s Dirichlet duality theory to the domains with corners. In this paper, we study the Lagrangian geodesic equation based on a different method. In particular, we consider the approximations of Lagrangian geodesics that pass from the degenerate Lagrangian geodesic equation to the special Lagrangian equation. Moreover, to avoid the problem of dealing with the domains with corners, we consider the exact Lagrangian graphs which are periodic perturbations of planes. In this case, one can reduce the Lagrangian geodesic equation over a cylinder. Then we solve the special Lagrangian equation by using the classical continuity method and establish some uniform estimates. Finally, by passing to the limit, we will show that two Lagrangian graphs with large Lagrange phases can be connected by Lipschitz continuous weak geodesics.
