Mean Curvature Vector Field in Calabi-Yau with Boundary

Question

It is well-known that the mean curvature vector field $H$ of a Lagrangian submanifold in a Calabi-Yau manifold $M$ is given by $H=J\nabla \theta$ where $\theta$ is the phase. I am interested in the case where either $M$ is not necessarily compact or has boundary. Does the equality hold in these cases as well? Any help would be appreciated.