  We introduce new boundary conditions for differential forms on symplectic
manifolds with boundary. These boundary conditions, dependent on the symplectic
structure, allows us to write down elliptic boundary value problems for both
second-order and fourth-order symplectic Laplacians and establish Hodge
theories for the cohomologies of primitive forms on manifolds with boundary. We
further use these boundary conditions to define a relative version of the
primitive cohomologies and to relate primitive cohomologies with Lefschetz maps
on manifolds with boundary. As we show, these cohomologies of primitive forms
can distinguish certain K\"ahler structures of K\"ahler manifolds with
boundary.
