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 Kahler structures of Kahler manifolds
with boundary.