We consider Landau-Ginzburg (LG) models with boundary conditions preserving A-type N = 2 supersymmetry. We show the equivalence of a linear class of boundary conditions in the LG model to a particular class of boundary states in the corresponding CFT by an explicit computation of the open-string Witten index in the LG model. We extend the linear class of boundary conditions to general non-linear boundary conditions and determine their consistency with A-type N = 2 supersymmetry. This enables us to provide a microscopic description of special lagrangian submanifolds in ℂn due to Harvey and Lawson. We generalise this construction to the case of hypersurfaces in ℙn. We find that the boundary conditions must necessarily have vanishing Poisson bracket with the combination (W(φ) - W̄(φ̄)), where W(φ) is the appropriate superpotential for the hypersurface. An interesting application considered is the T3 supersymmetric cycle of the quintic in the large complex structure limit.