Hodge theory on compact symplectic manifolds with boundary

We study Hodge theory for symplectic Laplacians on compact symplectic manifolds with bound- ary. These Laplacians are novel as they can be associated with symplectic cohomologies and be of fourth-order. We prove various Hodge decompositions and use them to obtain the isomorphisms between the cohomologies and the spaces of harmonic fields with certain prescribed boundary con- ditions. In order to establish Hodge theory in the non-vanishing boundary case, we are required to introduce new concepts such as the J-Dirichlet boundary condition and the J-Neumann boundary condition. When the boundary is of contact type, these conditions are closely related to the Reeb vector field. Another application of our results is to solve boundary value problems of differential forms.